February 1, 2008
LBNL-40356 UCB-PTH-97/28
Quantum Groups, Roots of Unity and
arXiv:hep-th/9705211v1 28 May 1997
Particles on quantized Anti–de Sitter Space 1
Harold Steinacker2
Department of Physics University of California, Berkeley, California 94720 and Theoretical Physics Group Ernest Orlando Lawrence Berkeley National Laboratory University of California, Berkeley, California 94720
A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics.
Committee in charge: Professor Bruno Zumino, Chair Professor Korkut Bardakci Professor Nicolai Y. Reshetikhin 1
This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-9514797. 2
email address:
[email protected]
Abstract Quantum groups in general and the quantum Anti–de Sitter group Uq (so(2, 3)) in particular are studied from the point of view of quantum field theory. We show that if q is a suitable root of unity, there exist finite–dimensional, unitary representations corresponding to essentially all the classical one–particle representations with (half)integer spin, with the same structure at low energies as in the classical case. In the massless case for spin ≥ 1, the ”naive” representations are unitarizable only after factoring out a subspace of ”pure gauges”, as classically. Unitary many–particle
representations are defined, with the correct classical limit. Furthermore, we identify a remarkable element Q in the center of Uq (g), which plays the role of a BRST operator in the case of Uq (so(2, 3)) at roots of unity, for any spin ≥ 1. The associated ghosts are an intrinsic part of the indecomposable representations. We show
how to define an involution on algebras of creation and anihilation operators at roots of unity, in an example corresponding to non–identical particles. It is shown how nonabelian gauge fields appear naturally in this framework, without having to define connections on fiber bundles. Integration on Quantum Euclidean space and sphere and on Anti–de Sitter space is studied as well. We give a conjecture how Q can be used in general to analyze the structure of indecomposable representations, and to define a new, completely reducible associative (tensor) product of representations at roots of unity, which generalizes the standard ”truncated” tensor product as well as our many–particle representations.
Disclaimer
This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or The Regents of the University of California.
Lawrence Berkeley Laboratory is an equal opportunity employer.
iii
Acknowledgements This thesis would not exist without the help of many people. First and foremost, it is a pleasure to thank my advisor Professor Bruno Zumino, who introduced me to this subject, for his kindness and support, for sharing his amazing intuition and insights, and especially for creating an atmosphere which is productive and pleasant at the same time. I consider myself very lucky having been his student. I would also like to thank my colleagues Chong–Sun Chu, Pei-Ming Ho and Bogdan Morariu for their friendship, and for uncountable stimulating discussions. The same goes for my fellow graduate students Nima Arkani–Hamed, Luis Bernardo, Kamran Saririan and Zheng Yin, who contributed much to the unique environment in Berkeley. I also want to thank Paolo Aschieri, Chris Chryssomalakos, Joanne Cohn, Julia Kempe, Michael Schlieker, Peter Schupp, Paul Watts and Miriam Yee, for interesting conversations and a good time. Needless to say, I learned a lot from many Professors in Berkeley, notably Korkut Bardakci and Nicolai Reshetikhin who were also on my thesis committee, as well as Mary K. Gaillard, among others. Professor Reshetikhin was able to explain to me some of the more mysterious aspect of the subject. Special thanks go to Laura Scott and Anne Takizawa, who made the bureaucratic matters on campus almost a pleasure, as well as their counterparts Barbara Gordon, Mary Kihanya and Louanne Neumann at LBL. Generally, I am grateful for having had the opportunity to study in Berkeley, which would not have been possible without the help of Professors Josef Rothleitner and Dietmar Kuhn in Innsbruck, and Gerald B. Arnold in Notre Dame. Last but by no means least, I thank my parents Christoph and Irmgard Steinacker for their support, faith and confidence. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-9514797.
1
Contents Introduction
1
1 Quantum Groups
5
1.1
1.2
Hopf Algebras and Quantum Groups . . . . . . . . . . . . . . . . . .
5
1.1.1
Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.1.2
Uq (g) and Quasitriangular Hopf Algebras . . . . . . . . . . . .
6
1.1.3
F un(Gq ) and dually paired Hopf Algebras . . . . . . . . . . .
9
1.1.4
More Properties of Uq (g) . . . . . . . . . . . . . . . . . . . . .
12
1.1.5
Reality Structures . . . . . . . . . . . . . . . . . . . . . . . . .
15
Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.2.1
18
1.2.2
Invariant Forms, Verma Modules and Unitary Representations ˆ Matrix and Centralizer Algebra . . . . . . . . . . . . . . . R–
1.2.3
Aspects of Representation Theory at Roots of Unity . . . . . .
23
2 Quantum Spaces associated to Quantum Groups 2.1
2.2
2.3
Definitions and Examples
22
27
. . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.1
Actions and Coactions . . . . . . . . . . . . . . . . . . . . . .
27
2.1.2
Quantum Spaces and Calculus as Comodule Algebras . . . . .
28
Integration on Quantum Euclidean Space and Sphere . . . . . . . . .
31
2.2.1
32
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . SqN −1
2.2.2
Integral on the Quantum Sphere
. . . . . . . . . . . . .
33
2.2.3
Integral over Quantum Euclidean Space
. . . . . . . . . . . .
38
2.2.4
Integration of Forms . . . . . . . . . . . . . . . . . . . . . . .
40
Quantum Anti–de Sitter Space . . . . . . . . . . . . . . . . . . . . . .
44
2.3.1
Definition and Basic Properties . . . . . . . . . . . . . . . . .
45
2.3.2
Commutation Relations and Length Scale . . . . . . . . . . .
47
2
3 The Anti–de Sitter Group and its Unitary Representations 3.1
3.2
The Classical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.1.1
SO(2, 3) and SO(2, 1) . . . . . . . . . . . . . . . . . . . . . .
49
3.1.2
Unitary Representations, Massless Particles and BRST from a Group Theoretic Point of View . . . . . . . . . . . . . . . . .
51
The Quantum Anti–de Sitter Group at Roots of Unity . . . . . . . .
55
3.2.1
Unitary Representations of SOq (2, 1) . . . . . . . . . . . . . .
56
3.2.2
Tensor Product and Many–Particle Representations of SOq (2, 1) 59
3.2.3
Unitary Representations of SOq (5) . . . . . . . . . . . . . . .
61
3.2.4
Unitary Representations of SOq (2, 3) . . . . . . . . . . . . . .
68
3.2.5
Tensor Product and Many–Particle Representations of SOq (2, 3) 71
3.2.6
Massless Particles, Indecomposable Representations and BRST for SOq (2, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Operator Algebra and BRST Structure at Roots of Unity 4.1
49
72 76
Indecomposable Representations and BRST Operator . . . . . . . . .
76
4.1.1
A Conjecture on BRST and Complete Reducibility . . . . . .
80
An Inner Product on G0 . . . . . . . . . . . . . . . . . . . . . . . . .
81
4.2.1
Hilbert Space for the Quantum Anti–de Sitter group . . . . .
83
4.3
The Universal Weyl Element w . . . . . . . . . . . . . . . . . . . . .
84
4.4
An Algebra of Creation and Anihilation Operators . . . . . . . . . . .
88
4.4.1
On Quantum Fields and Lagrangians . . . . . . . . . . . . . .
91
4.4.2
Nonabelian Gauge Fields from Quantum AdS Space . . . . . .
94
4.2
5 Conclusion
97
A Consistency of Involution, Inner Product
99
B On the Weyl Element w
102
3
Introduction The topic of this thesis is the study of quantum groups and quantum spaces from the point of view of Quantum Field Theory. The motivation behind such an endeavour is easy to see. Quantum field theory (QFT) is a highly successful theory of elementary particles, with an embarrassing ”fault”: except for some special cases, it cannot be defined without some sort of a regularization of the underlying space, which at present is little more than a recipe to calculate divergent integrals. Physically, it is in fact expected that space–time will not behave like a classical manifold below the Planck scale, where quantum gravity presumably modifies its structure. The hope is that this somehow provides a regularization for QFT. From a mathematical point of view, it also seems that there should exist some rigorous theory behind such ”naive” quantum field theories, given the rich mathematical structures apparently emerging from them. And of course, it would be highly desirable to put the physically relevant quantum field theories on a firm theoretical basis. In view of this, it seems very natural that a consistent theory of elementary particles should not be based on concepts of classical geometry, but rather on some kind of ”fuzzy”, or ”quantized” space–time. With very little experimental guidance, finding the correct description may seem rather hopeless. The approach we will pursue here relies heavily on mathematical guidance, given the ”unreasonable usefulness of mathematics in physics” (Wigner). With the development of Non–Commutative Geometry in recent years [8], a possible candidate for a new framework has emerged. It is a generalization of the (rather old) idea that a manifold M can be described by the algebra of functions F un(M)
on it. In Non–Commutative Geometry, one considers instead some non–commutative
algebras replacing F un(M), with sufficiently rich additional structures. A ”quantum deformation” or simply ”deformation” of a classical manifold is essentially a (noncommutative) algebra with a deformation parameter h, such that the classical algebra 1
of functions on the manifold is obtained in the limit h → 0.
This idea is in fact very familiar to physicists: Quantum Mechanics can be viewed
as a noncommutative geometry on phase space, and the Planck constant plays the role of the deformation parameter. This example also shows that while the limit h → 0 may be smooth in some sense, the physical interpretation may be very different. Furthermore if h is dimensional, it is expected that the ”quantum” case should behave classically at large enough scales. This is a new and vast field of research, and to make any progress, one clearly has to choose a particular approach. A simple example of a theory of elementary particles on a noncommutative space was proposed by Connes [9] and Connes & Lott [10]. It is based on the space M × ZZ2 , where M is ordinary Minkowski space and ZZ2
is considered as a noncommutative space with a connection, which can be interpreted as a Higgs field. This leads to a new approach to the standard model. Fr¨ohlich and collaborators [5] introduced gravity in this context. Incidentally, it has been pointed out that string theory, a candidate for a theory
underlying quantum field theory including gravity, seems to predict some noncommutativity of certain coordinate algebras [60]. Other recent developments [3] also suggest some relevance of Non-Commutative geometry to M–theory or string theory, which is traditionally formulated in the language of classical geometry. Of course one would really like to consider truly noncommutative spaces. There have been many approaches to ”quantize” physically relevant spaces like Minkowski space, fiber bundles, and many others. While many interesting examples have been found, a clear guideline is missing. At this point, we want to emphasise (without the need to do so) the importance of Lie groups in elementary particle physics, notably the Poincare group which dictates the behaviour of free particles, and internal symmetry groups which may strongly constrain their interactions. Quantum groups are remarkable examples of Noncommmutative Geometry, since they can be viewed as deformations of classical Lie groups resp. their manifolds. Their mathematical structure is well studied and even richer than that of classical Lie groups. They depend on a deformation parameter q = eh , where h = 0 corresponds to the classical case. Furthermore, they act naturally on associated quantum spaces. Thus it seems that quantum groups, which combine the features of both Lie groups and Non–Commutative Geometry in an analytic way, should be a powerful guide towards a realization of the above ideas. In this thesis, we want to follow this 2
approach and see where it leads to. It is fair to say that quantum groups are analytic ”deformations” of classical groups, for ”generic” q. However when q is a root of unity, their structure is in many ways very different, and one is facing a truly new and very rich mathematical object. One of the main points we want to make is that the root of unity case seems to be the most interesting one from a QFT point of view, beyond its known relevance to Conformal Field Theory [1, 40]. We will mainly study the Anti–de Sitter group SOq (2, 3) and its representation theory, which will play the role of the Poincare group. This choice is vindicated by its simplicity and the wealth of interesting features found, which constitute the main part of this thesis. In the classical case, the Poincare group can be obtained from SO(2, 3) by a contraction; however we will not do a corresponding contraction in the quantum case [36], since our main results would all break down. This thesis is organized as follows. Chapter 1 is a brief, general introduction to quantum groups and their representation theory, with emphasis on those aspects which will be important later. Whenever possible, a short explanation is included on how these facts can be obtained. We will mainly work with the ”quantized universal enveloping algebra” Uq (g). Most of this chapter is well known, but it also contains some new results and definitions. In chapter 2, we consider quantum spaces associated to quantum groups, and study integration on quantum Euclidean space, sphere, and on quantum Anti–de Sitter (AdS) space. We point out that quantum AdS space has an intrinsic length scale, above which it looks like a classical manifold. Chapter 3 starts with a brief review of the unitary representations of the classical AdS group corresponding to elementary particles, as well as a discussion of massless particles, (abelian) gauge theories and BRST from a group theoretic point of view. We then study these issues for the quantum AdS group. In particular, we show that for suitable roots of of unity q, there are finite–dimensional, unitary representations corresponding to all the classical ones, with the same structure at low energies3 . In the massless case for spin ≥ 1, the ”naive” representations contain a subspace of ”pure gauges” which must be factored out to get unitary, irreducible representations,
as classically. A definition of unitary many–particle representations is given. Furthermore, we identify a remarkable element Q of the center of Uq (g) which plays the role of a (abelian) BRST operator in the case of the AdS group at roots of unity, for 3
For the singleton representations, this was already shown in [12].
3
any spin ≥ 1. The corresponding ghosts are an intrinsic part of the indecomposable
representations.
In chapter 4, we give a conjecture that Q can be used for any group to understand the structure of the tensor product at roots of unity, and to define a new, associative (tensor) product of irreducible representations at roots of unity, which generalizes the well–known ”truncated” tensor product used in conformal field theory, as well as the many–particle representations mentioned above. We then show how one can define an involution on algebras of creation and anihilation operators, for q a root of unity; lacking a symmetrization postulate at present, we have to work with a version corresponding to non–identical particles. It is shown how all this might be used towards constructing a quantum field theory. The main missing piece to achieve this goal is a way to define identical particles, i.e. a symmetrization postulate. Finally, we point out that nonabelian gauge fields appear naturally in this framework, without having to define something like connections on fiber bundles.
4
Chapter 1 Quantum Groups 1.1
Hopf Algebras and Quantum Groups
In this thesis, we will be concerned with the representation theory of quantum groups in the Drinfeld–Jimbo formulation, which is a certain Hopf algebra with additional structure to be explained below. The most economical approach to this goal would be to start with these given mathematical objects, and study its properties from the point of view we have in mind. However a reader who is not very familiar with quantum groups would be left in the dark wondering where all this comes from, and quite possibly develop some misconceptions. Therefore we first give a brief review of the underlying mathematical structure. For more details, the reader is referred to [15, 18] or a number of existing reviews, such as [6, 25]. There are at least two ways to introduce quantum groups. One is to consider the universal enveloping algebra of a simple Lie group, and discover a new structure on it, namely that of a quasitriangular Hopf algebra. The other, more geometric approach is to ”quantize” the space of functions on a (compact) Lie group, which turns out to have a remarkable Poisson–structure. These two approaches are dual to each other, and they originated in the study of certain integrable models.
1.1.1
Hopf Algebras
The mathematical language to describe both points of view is that of a Hopf algebra. The most familiar example of a Hopf algebra A is the space of functions F un(G) on
a (compact) Lie group G. This is a commutative algebra by pointwise multiplication,
but this has nothing to do with the group structure. The group multiplication is 5
encoded in A as a coproduct, which is a map ∆ : A → A ⊗ A, where (∆(f ))(x, y) =
f (x · y) for f ∈ F un(G). The inverse is encoded as antipode S : A → A where
(Sf )(x) = f (x−1 ) in the case of F un(G), and the unit element e ∈ G becomes the
| , with ǫ(f ) = f (e) for F un(G). In this way, all the structure of counit ǫ : A → C
G has been encoded in F un(G). In general, a Hopf algebra is an algebra A with
coproduct, antipode and counit and the following compatibility conditions: (∆ ⊗ id)∆(a) = (id ⊗ ∆)∆(a),
(coassociativity),
·(ǫ ⊗ id)∆(a) = ·(id ⊗ ǫ)∆(a) = a,
(counit),
·(S ⊗ 1)∆(a) = ·(id ⊗ S)∆(a) = 1ǫ(a),
(coinverse),
∆(ab) = ∆(a)∆(b), ǫ(ab) = ǫ(a)ǫ(b), ∆(1) = 1 ⊗ 1,
(1.1) (1.2) (1.3) (1.4)
and
(1.5)
ǫ(1) = 1,
(1.6)
for all a, b ∈ A. This implies S(ab) = S(b)S(a),
(antihomomorphism),
S(1) = 1,
(1.7) (1.8)
∆(S(a)) = τ (S ⊗ S)∆(a),
with τ (a ⊗ b) ≡ b ⊗ a,
ǫ(S(a)) = ǫ(a).
(1.9) (1.10)
We will use Sweedler’s [54] notation for the coproduct: ∆(a) ≡ a(1) ⊗ a(2)
(summation is understood).
(1.11)
A is not required to be commutative, and in general it is non-cocommutative, i.e.
∆′ ≡ τ ◦ ∆ 6= ∆.
1.1.2
Uq (g) and Quasitriangular Hopf Algebras
The fastest way to introduce quantum groups is to simply write down a certain deformation of the universal enveloping algebra of a simple Lie algebra g in a Chevalley basis with a complex parameter q, and study its properties. We will mainly work in this framework, which was introduced by Drinfeld [15] and Jimbo [27]. (αi ,αj ) | and A Let q ∈ C ij = 2 (αj ,αj ) be the Cartan matrix of a classical simple Lie algebra
g of rank r, where (, ) is the Killing form and {αi , 6
i = 1, ..., r} are the simple roots.
Then the quantized universal enveloping algebra U ≡ Uq (g) is the Hopf algebra with
generators {Xi± , Hi; h
[Hi , Hj ] = 0
Hi , Xj±
h
i = 1, ..., r} and relations [18, 27, 15]
Xi+ , Xj−
i
i
(1.12)
= ±Aji Xj± ,
(1.13)
= δi,j
(1.14)
1−Aji
X
q di Hi − q −di Hi = δi,j [Hi ]qi q di − q −di
1 − Aji (Xi± )k Xj± (Xi± )1−Aji −k = 0, k q
k=0
i 6= j
(1.15)
i
qi = q di ,
where di = (αi , αi )/2,
qin −qi−n qi −qi−1
[n]qi =
and
n [n]qi ! . = [m]qi ![n − m]qi ! m
(1.16)
qi
The last of (1.15) are the deformed Serre relations. As algebra, it can be shown [16] that if q is considered as a formal variable, U is essentially1 the same as the classical,
undeformed enveloping algebra with a formal variable q and a different choice of generators. However it is not equivalent as Hopf algebra: the comultiplication on U is defined by
∆(Hi ) = Hi ⊗ 1 + 1 ⊗ Hi
∆(Xi± ) = Xi± ⊗ q di Hi /2 + q −di Hi /2 ⊗ Xi± ,
(1.17)
and antipode and counit are S(Hi ) = −Hi ,
S(Xi+ ) = −q di Xi+ ,
ǫ(Hi ) = ǫ(Xi± ) = 0.
S(Xi− ) = −q −di Xi− , (1.18)
The classical case is obtained by taking q = 1. The consistency of this definition can be checked explicitely. The Cartan–Weyl involution is defined as θ(Xi± ) = Xi∓ , 1
without going into mathematical detais here
7
θ(Hi ) = Hi ,
(1.19)
extended as a linear anti–algebra map (involution). In particular, θ(q) = q for any | . It is obviously consistent with the algebra, and one can check that q∈C
(θ ⊗ θ)∆(x) = ∆(θ(x)),
S(θ(x)) = θ(S −1 (x)).
(1.20) (1.21)
The conventions we use are those of [18] except for a replacement q → q −1 for
reasons explained in the next section, and agree up to normalization (see below) with those of [15, 33]. They differ from e.g. [6] by some redefinitions. In the mathematical literature, usually a rational version of the above algebra, i.e. using q di Hi instead
of Hi is considered. Since we are mainly interested in specific representations, we prefer to work with Hi . Furthermore, if q is a root of unity, one has to specify if one includes the ”divided powers” (Xi± )(k) =
(Xi± )k [k]qi !
(”restricted specialization”) or not
(”unrestricted specialization”); the representation theory is quite different in these cases. We will mostly work in the unrestricted specialization, however since we are really only interested in certain representations, it will become clear from the context what is appropriate. Often the following operators are often more useful: hi = di Hi ,
e±i =
q
[di ]Xi± ,
(1.22)
Then the first two relations in (1.15) become [hi , e±j ] = ±(αi , αj )e±j ,
(1.23)
[ei , e−j ] = δi,j [hi ]q .
(1.24)
In order to have the standard Physics normalization for angular momenta, the normalization of the Killing form will be chosen so that the short roots have length di = 12 , i.e. (αi , αi ) = 1, and the long roots have length 1. In any case, a rescaling of the Killing form can be absorbed by a redefinition of q. One could also define another Hopf algebra with reversed coproduct ∆′ (x) = τ ◦ ∆(x), and S −1 instead of S. However this is essentially the same. The reason is
that U has the very important property of being quasitriangular, i.e. there exists a universal R ∈ U ⊗ U with the following properties:
(∆ ⊗ id)R = R13 R23 ,
(1.25)
(id ⊗ ∆)R = R13 R12
(1.26)
∆′ (x) = R∆(x)R−1 8
(1.27)
for any x ∈ U, where lower indices denote the position of the components of R in
the tensor product algebra U ⊗ U ⊗ U : if R ≡ ai ⊗ bi (summation is understood),
then e.g. R13 ≡ ai ⊗ 1 ⊗ bi . By considering (∆′ ⊗ id)R = R23 R13 , one obtains the Quantum Yang–Baxter equation
R12 R13 R23 = R23 R13 R12 .
(1.28)
Furthermore, the following properties are a consequence of (1.25) to (1.27): (S ⊗ id)R = R−1 ,
(id ⊗ S)R−1 = R,
(ǫ ⊗ id)R = (id ⊗ ǫ)R = 1.
(1.29) (1.30) (1.31)
The construction of R and the proof of the relations (1.25) to (1.27) is based on the
so–called quantum double construction due to Drinfeld [15]. It turns out that the Borel subalgebras B− and B+ , generated by {Hi , Xj− } and {Hi , Xj+ } respectively, are Hopf–subalgebras which are dually paired (see below). If {ai } is a basis of B− and
{bi } the dual basis of B+ , then R = ai ⊗ bi , after factoring out a copy of the Cartan subalgebra which has been counted twice. The relations (1.25) to (1.27) are then easy to see. This universal R is the essential feature of a quantum group, and we will make
extensive use of it. It incorporates the additional structure of Lie groups which is not used in the classical theory, namely the existence of a certain Poisson structure compatible with the group structure. Furthermore, all this is combined into objects
which are holomorphic in q. Therefore one should expect that there is a lot to say about this rich structure.
1.1.3
F un(Gq ) and dually paired Hopf Algebras
Before studying U any further, let us now sketch the second approach to quantum
groups; for a general review see [6]. It is based on the observation that any (compact)
Lie group G with Lie algebra g is actually a (coboundary) Poisson–Lie group, i.e. there is a particular Poisson structure on the group manifold which can be written in terms of a ”classical r-matrix” r ∈ g ⊗ g, and enjoys certain compatibility conditions.
r can again be obtained from a ”double construction” [15]; it is not given by the structure constants of G, it is truly an additional structure on G. Now as in Quantum Mechanics on a phase space, this Poisson structure can be quantized, giving rise to a 9
non–commutative algebra F un(Gq ) which replaces the commutative algebra F un(G). If one writes q = eh , h plays the role of the Planck constant, and the classical case F un(G) is obtained in the limit h → 0, i.e. q → 1. This is the origin of the name
”Quantum group”, and even though this quantization procedure may be formal, the final result is known to exist. Upon this quantization, the ”classical r-matrix” r turns into the universal R ∈ U ⊗ U.
These two approaches are in fact dual to each other. This means that Uq−1 ≡
Uq−1 (g) and A = F un(Gq ) are dually paired Hopf algebras (notice the replacement
q → q −1 ). In general, two Hopf algebras U and A are said to be dually paired if there | , such that: exists a non-degenerate inner product < , >: U ⊗ A → C
< xy, a > = < x ⊗ y, ∆(a) >≡< x, a(1) >< y, a(2) >, < x, ab > = < ∆(x), a ⊗ b >, < S(x), a > = < x, S(a) >, < x, 1 > = ǫ(x),
and
< 1, a >= ǫ(a),
(1.32)
for all x, y ∈ U and a, b ∈ A.
The algebra F un(Gq ) can be written down explicitely [18] if it is written as
a pseudo matrix group [61], generated by the elements of a N × N matrix A ≡
(Ai j )i,j=1...N ∈ MN (F un(Gq ))2 . The coproduct on F un(Gq ) is defined as classically, ˙ ∆A = A⊗A,
i.e. ∆(Ai j ) = Ai k ⊗ Ak j ,
(1.33)
and S(Ai j ) = (A−1 )i j , ǫ(Ai j ) = δ i j . Now if < , > is a dual pairing of Uq−1 with F un(Gq ), then this implies that π i j ≡< . , Ai j > is a representation of Uq−1 , i.e. | , π i j : Uq−1 → C
π i j (xy) =
P
k
π i k (x)π k j (y),
∀x, y ∈ Uq−1 ;
(1.34)
we will say much more about representations in a later section. In this representation, the universal R ∈ Uq−1 ⊗ Uq−1 gives the numerical R-matrix: < R, Ai k ⊗ Aj l >= Rij kl .
(1.35)
Now the definition of a dual pairing (1.32) and (1.27) imply [15, 18]
2
< x, Aj s Ai r > = < ∆x, Aj s ⊗ Ai r >
This corresponds to GLq (N ) unless there are explicit or implicit restrictions on the matrix
elements of A.
10
= < τ ◦ ∆x, Ai r ⊗ Aj s >
= < R(∆x)R−1 , Ai r ⊗ Aj s >
= < x, Rij kl Ak m Al n (R−1 )mn rs >,
(1.36)
for any x ∈ Uq−1 , i.e. the matrix elements of A satisfy the commutation relations Rij kl Ak m Al n = Aj s Ai r Rrs mn ,
(1.37)
which can be written more compactly in tensor product notation as follows: R12 A1 A2 = A2 A1 R12 ; R12 = (π1 ⊗ π2 )(R) ≡ < R, A1 ⊗ A2 > .
(1.38) (1.39)
Starting from this formalism, one can introduce differential forms etc. and study the noncomutative differential geometry of quantum groups, see [62, 64, 51, 57]. So far, | we are considering all algebras over C without any reality structure, which we will
discuss below. Now one can recast the commutation relations of Uq−1 (g) into a more compact form [18] . For a representation π, define matrices L+ ≡ (id ⊗ π)(R), π
SL− ≡ (π ⊗ id)(R), π
L− ≡ (π ⊗ id)(R−1 ). π
(1.40)
Then the commutation relations for these matrices follow from the quantum YangBaxter equation, e.g. 0 = (id ⊗ π ⊗ π)(R23 R13 R12 − R12 R13 R23 ) + + + = R12 L+ 2 L1 − L1 L2 R12
(1.41) (1.42)
and similarly − − R12 L− = L− 2 L1 1 L2 R12 ,
(1.43)
− + R12 L+ = L− 2 L1 1 L2 R12 .
(1.44)
˙ ±, ∆L± = L± ⊗L
(1.45)
The coproduct is now and ǫ(L± ) = I, S(L± ) = (L± )−1 . The Xi± can be extracted from the upper resp. lower triangular matrices L± [18]. 11
One can also turn the logic around and show that there exists a dual pairing between Uq−1 (g) and the Hopf algebra defined by (1.38) and (1.33) (with some suitable additional constraints depending on the group, cp. [41]), which can be seen to be a quantization of F un(G).
1.1.4
More Properties of Uq (g)
Let us describe U in more detail. In the classical case, the Weyl group W acting on
weight space by the reflections σi along the simple roots can be ”lifted” to an action of the braid group3 with generators Ti on representations of g, in particular on g itself with the adjoint representation. The relations of the braid group are (Ti Tj )mij = 1 if (σi σj )mij = 1, but the square of Ti is not required to be 1 any more. The same can be done for U [37]: there exist algebra automorphisms of U defined as Ti (Hj ) = Hj − Aij Hi , Ti (Xj+ )
=
−Aji
X
Ti Xi+ = −Xi− qiHi ,
(−1)r−Aji qi−r (Xi+ )(−Aji −r) Xj+ (Xi+ )(r)
(1.46)
r=0
where (Xi± )(k) =
(Xi± )k , [k]qi !
and similarly for lowering operators4 . If ω = σi1 ...σiN is a
reduced expression for the longest element of the Weyl group, then {β1 = αi1 , β2 =
σi1 (αi2 ), . . . , βN = σi1 ...σiN−1 (αiN )} is an ordered set of positive roots. Now one can define root vectors of U as Xβ±r = Ti1 ...Tir−1 (Xi±r ). This can be used to obtain a
Poincare–Birkhoff–Witt basis of U = U − U 0 U + where U ± is generated by the Xi± and U 0 by the Hi : for k = (k1 , ..., kN ), let Xk+ = (Xβ+N )kN ...(Xβ+1 )k1 and similarly for Xk− .
Then the Xk+ form a P.B.W. basis of U + , and similarly for U − [38].
Using this, one can find explicit formulas for R = R(q) [32, 33]. They are some-
what complicated however, and all we need for now is the following form: R=q
P
(α−1 )ij hi ⊗ hj
X
˜+ ⊗ X ˜ −′ ck,k′ (q)X k k
(1.47) 1
˜ ± is defined like X ± with Xi± replaced by X ˜ i± ≡ qi± 2 Hi Xi± , where5 (α)ij = (αi , αj ), X k k
| are rational functions of q. Furthermore, the coefficients in (1.47) are and ck,k′ (q) ∈ C
uniquely determined by the properties (1.25) to (1.27) [16, 32]. Using this, is easy to 3
this has nothing to do with the representations of the braid group obtained from R. the Ti can actually be implemented as ωi (...)ωi−1 in an extension of U, see [33, 34]; we will come back to that. 4
5
Since we will work with ”nice” representations, R converges.
12
see that R(q −1 ) = R−1 (q),
(1.48)
(θ ⊗ θ)R12 = R21 .
(1.49)
For the case of Uq (sl(2)), the explicit form is R(q) = q
1 H 2
⊗H
X
q
1 l(l−1) 2
l≥0
1 (1 − q −2 )l 1 H + l (q 2 X ) ⊗(q − 2 H X − )l [l]q !
(1.50)
where we have set d = (α, α)/2 = 1, i.e. qi = q.
It is easy to check that the square of the antipode is an inner automorphism S 2 (x) = q 2˜ρ xq −2˜ρ , where ρ˜ =
P
α>0
hα and hα =
P
ni hi if α =
P
(1.51)
ni αi [15]. As shown in [15], there is
another element in U with that property, namely u ≡ SR2 R1 . Therefore v ≡ (SR2 )R1 q −2˜ρ
(1.52)
commutes with any element in U and will be called Drinfeld–Casimir. It satisfies ∆(v) = (R21 R12 )−1 v ⊗ v,
(1.53)
v −1 = q 2˜ρ R2 S 2 (R1 ),
(1.54)
S(v) = v
(1.55)
where R12 = R and R21 = τ ◦ R. Furthermore, it is easy to check from (1.49) that θ(v) = v.
(1.56)
The value of v on a highest weight representation was first determined in [49], and can be obtained easily from (1.47): if wλ is a h.w. vector with X + · wλ = 0 and hi · wλ = (λ, αi)wλ (see section 1.2), then
v · wλ = q −cλ wλ ,
(1.57)
where cλ = (λ, λ + 2ρ) is the value of the classical quadratic Casimir on wλ . l
Later we will need analogs of R for U ⊗ l with l > 2. If ∆(l) (x) ∈ U ⊗ is the
(unique) l – fold coproduct of x ∈ U, let (a)
(a)
R12...l ≡ (R12...(l−1) ⊗ 1)(∆(l−1) ⊗ id)R12 ,
(1.58)
R12...l ≡ (1 ⊗ R12...(l−1) )(id ⊗ ∆(l−1) )R12
(1.59)
(b)
(b)
for l > 2. They have similar properties as R: 13
Lemma 1.1.1 (a)
(b)
R12...l = R12...l ≡ R12...l ,
(1.60)
and ∆′(l) (x) = R12...l ∆(l) (x)R−1 12...l ,
R12...l (q −1 ) = R−1 12...l (q), (θ ⊗ . . . ⊗ θ)R12...l = Rl...21.
Proof
(1.61) (1.62) (1.63)
(1.60) follows by straightforward induction: For l = 3, it reduces to the
Yang–Baxter equation. By the induction assumption, we have (a)
R12...l = (R12...(l−1) ⊗ 1)(id ⊗ ∆(l−2) ⊗ id)(∆ ⊗ 1)R12 (b)
= (R12...(l−1) ⊗ 1)(id ⊗ ∆(l−2) ⊗ id)R13 R23 = (1 ⊗ R12...(l−2) ⊗ 1)(id ⊗ ∆(l−2) ⊗ id)R12 R13 R23 .
(1.64)
Similarly, (b)
(a)
R12...l = (1 ⊗ R12...(l−1) )(id ⊗ ∆(l−2) ⊗ id)(id ⊗ ∆)R12 = (1 ⊗ R12...(l−2) ⊗ 1)(id ⊗ ∆(l−2) ⊗ id)R23 R13 R12 ,
(1.65)
and (1.60) follows using the Yang–Baxter equation. Applying (∆(l−1) ⊗ id) to (1.27), one obtains (∆(l−1) x(2) ) ⊗ x(1) = ((∆(l−1) ⊗ id)R12 )∆(l) (x)((∆(l−1) ⊗ id)R−1 12 ), and by
induction and using (1.60) it follows
(∆′(l−1) x(2) ) ⊗ x(1) = (R12...(l−1) ⊗ 1)((∆(l−1) ⊗ id)R12 )∆(l) (x) ·
(1.66)
−1 ((∆(l−1) ⊗ id)R−1 12 )(R12...(l−1) ⊗ 1) (a)
(a)
= R12...l ∆(l) (x)(R12...l )−1 ,
(1.67)
which shows (1.61). For illustration let us also show (1.63). From (1.58), one gets by induction (a)
(θ ⊗ . . . ⊗ θ)R12...l = ((∆(l−1) ⊗ id)R21 )(R(l−1)...21 ⊗ 1)
= (R(l−1)...21 ⊗ 1)((∆′(l−1) ⊗ id)R21 ) (b)
= Rl...21 = Rl...21 , using the flipped (1.61) and (1.60). Similarly one can see (1.62). ⊔ ⊓ All this will become more obvious in section 4.3. 14
(1.68)
1.1.5
Reality Structures
| , in particular U is the universal enveloping So far we considered all algebras over C
algebra of a complex Lie algebra. If one wants to consider e.g. SOq (5) ≡ Uq (so(5; IR))
or SOq (2, 3) ≡ Uq (so(2, 3; IR)), one has to introduce a star–structure, an antilinear
involution x, as classically. This star–structure must be such that if x ∈ U is a group element for q = 1, e.g. x = exp(y) for y an element in the Lie algebra, then x = x−1 ,
i.e. x becomes the adjoint of x in a unitary representation of U (this will be considered in detail below).
There are several possible star–structures on U. One has to distinguish the cases
q ∈ IR and |q| = 1. If q ∈ IR, then a natural definition is xr ≡ θ(x∗ )
(1.69)
x ⊗ y r ≡ xr ⊗ y r ,
(1.70)
with r
∆(x) = ∆(xr )
(1.71)
r
and Sx = S −1 xr , which is a standard Hopf algebra star–structure (x∗ is the complex conjugate of x ∈ U, where the Xi± are considered real). Using the uniqueness of R (see the discussion below (1.47)), one can see that r
R12 = R21 = τ ◦ R.
(1.72)
If |q| = 1, a natural definition is xc ≡ θ(x∗ )
(1.73)
x ⊗ y c ≡ y c ⊗ xc
(1.74)
with c
∆(x) = ∆(xc )
(1.75)
c
and Sx = Sxc , which is a nonstandard Hopf algebra star–structure; notice that c
(a)
c
(b)
q c = q −1 . In this case, R = R−1 , and more generally R12...l = (R12...l )−1 with the obvious extention of (1.74) to several factors, thus c
R12...l = R−1 12...l using Lemma 1.1.1. 15
(1.76)
c
−1 2 −1 Furthermore, (SR2 )R1 = R−1 (see [6]), so 2 SR1 = R2 S R1 = (SR2 R1 )
vc = v −1 .
(1.77)
Both xr and xc correspond to the compact case, such as SOq (5); however we will mainly be interested in the case of |q| = 1. Having applications in QFT in mind, one might then be worried about (1.74). We will see in sections 4.4, 4.2 and to some extent
in section 1.2.1 how this can be used consistently with a many–particle interpretation. In the classical case, the coproduct on U is cocommutative, and it does not make any
difference whether its components are flipped by the reality structure or not.
Reality structures corresponding to noncompact groups can be obtained from c
x by conjugation with elements of the Cartan subalgebra. We will only consider star–structures of the form Xi+ = ±Xi− and Hi = Hi . For example, SOq (2, 1) is
| )) with H = H and X ± = −X ∓ , which can be realized as the algebra Uq (sl(2, C
x = (−1)−H/2 xc (−1)H/2 . Then for |q| = 1, again R = ((−1)−H/2 ⊗(−1)−H/2 )R−1 ·
((−1)H/2 ⊗(−1)H/2 ) = R−1 . The cases SOq (2, 1) and SOq (2, 3) will be considered in
much more detail below. We will only consider star–structures with R = R−1
(1.78)
for |q| = 1.
1.2
Representation Theory
We will only consider the representation theory of U. The main advantage of this
point of view is that the representation theory of U can be formulated in the familiar
language of ordinary semi–simple Lie algebras.
Let us first collect the basic definitions. We will (essentially) only consider finite– dimensional representations. A representation of U on a vectorspace V is a map
U → GL(V ) such that (xy) · v = x · (y · v) and 1 · v = v (sometimes we will
omit the · ). Then one can as usual diagonalize the Cartan subalgebra, and every
representation is a sum of weight spaces. A vector vλ has weight λ if hi vλ = (λ, αi)vλ . Then Xi± are rising and lowering operators as in the classical case, since (1.12) and (1.23) are undeformed. We will mainly (but not always) consider the case of integral weights, i.e. (λ, βi) ∈ di ZZ. Classically, all irreducible representations are highest
weight representations V with dominant integral highest weight λ, i.e. V = Uwλ and 16
(λ, αi ) > 0 for all simple roots αi . Finally, let Q = +
Q =
P
P
ZZαi be the root lattice and
ZZ+ αi where ZZ+ = {0, 1, 2, ...}. We will write
λ ≻ µ if λ − µ ∈ Q+ .
(1.79)
Given two representations V1 , V2 of a Hopf algebra, the tensor product representation is naturally defined as x · (v1 ⊗ v2 ) ≡ ∆(x)(v1 ⊗ v2 ) = (x(1) · v1 ) ⊗(x(2) · v2 ).
So far, we have not specified q at all. For the representation theory however, there
are two very different cases. One is if q is ”generic”, i.e. not a root of unity, and the other is if q is a root of unity. If q is generic, then the representation theory is essentially the same as in the classical case, in the sense that all the important theorems have a perfect analog. This is quite intuitive, since everything will be holomorphic in q, which as always is a very strong constraint. We will quote the main results in a moment. If q is a root of unity however, the representation theory changes completely, and essentially none of the classical theorems continue to hold. Roughly speaking this happens because poles resp. zeros occur in various quantities. While it is more complicated and therefore often discarded, this is the truly interesting case. The main objective of this thesis is to point out that many of its features seem to be very relevant to Quantum Field Theory, and not only to Conformal Field Theory. In any case, the root of unity case is not well enough understood, and deserves further study. Consider first the case of generic q. Then the basic results are as follows: • Any finite–dimensional representation of U is completely reducible, i.e. it decomposes into a direct sum of irreducible representations (irreps).
• The irreps are highest weight representations with dominant integral highest weight λ, and the representation space is the same as classically.
• The fusion rules are the same as classically. So the Weyl group acts on the weights of an irrep, and in fact a braid group action can be defined on any representation [39, 25, 33]. Complete reducibility was first proved by Rosso [50]. These results are not hard to understand. The first two would be obvious if one could apply the fact that as algebra, U = Uq (g) is nothing but the classical enveloping
algebra [17] (with a formal variable q however, and the correspondence may not be | ). Since similar issues will arise later, we want to explain here realized for a given q ∈ C
17
why a representation V of U which is irreducible for q = 1 can become reducible only
for q a root of unity. Such a representation must have a dominant integral highest
weight λ. If it contains a submodule at q0 with a highest weight vector with weight λ0 (which is dominant again by virtue of the Weyl group resp. braid group action), c
then the Drinfeld Casimir v must be the same on this submodule, so q0cλ = q0λ0 using (1.57). However λ ≻ λ0 , and we have Lemma 1.2.1 If λ, λ0 are dominant weights with λ ≻ λ0 , then cλ > cλ0 .
Proof
Notice first that cλ ≡ (λ, λ + 2ρ) = cσ˜i (λ) , where σ ˜i (λ) = λ −
(1.80)
(λ+ρ,βi ) βi di
is
the modified action of the Weyl reflection along any root βi with reflection center −ρ.
Since λ ≻ λ0 and both are dominant, λ0 is contained in the convex hull of λ and the σ ˜i (λ). But the Killing form is Euclidean and therefore convex, and the statement follows. ⊔ ⊓ Therefore if this V is not irreducible at q0 , q0 must be a phase, and in fact a root of unity (we assume q0 6= 0).
Complete reducibility can be understood using the concept of invariant sesquilin-
ear forms.
1.2.1
Invariant Forms, Verma Modules and Unitary Representations
A bilinear form ( , ) on a representation V is linear in both arguments, while a sesquilinear form is linear in the second argument and antilinear in the first. A bilinear form is called invariant if (u, x · v) = (θ(x) · u, v)
(1.81)
for u, v ∈ V ; this is sometimes called covariant [13]. This can be considered for any
| . q∈C
For q ∈ IR or |q| = 1, consider a star–structure as in section 1.1.5 and denote it
by x, so Xi± = ±Xi∓ and Hi = Hi . Then a sesquilinear form ( , ) is called invariant
if
(u, x · v) = (x · u, v) 18
(1.82)
for u, v ∈ V ; this is also sometimes called covariant. It is hermitian if (u, v)∗ = (v, u).
(1.83)
A hermitian sesquilinear form is called an inner product. Note that we always consider q to be a complex number, so q = q ∗ ; in the literature, q is often treated as an indeterminate, and our definitions agree with those of e.g. [13] only for |q| = 1, which
is the case we are mainly interested in. Finally, a representation V is unitary or
unitarizable if there exists a positive definite invariant inner product on V . Given a highest weight (h.w.) representation V (λ) with h.w. vector wλ , there is a unique invariant inner product ( , ) on V (λ) for q ∈ IR or |q| = 1, resp. an invariant | . Uniqueness is clear, since one can express symmetric bilinear form for any q ∈ C
any (U · wλ , U · wλ ) in terms of (wλ , U 0 · wλ ) or (U 0 · wλ , wλ ) using invariance and the
commutation relations. These two results agree and ( , ) is hermitian, because the star–structure is consistent with the commutation relations; notice that [hi ]q ∈ IR.
Thinking of applications in Quantum Mechanics, the importance of unitarity is
obvious. But invariant sesquilinear forms (or bilinearforms) are also very useful as technical tools, due to the following well–known observation: if a highest weight representation V (λ) is not irreducible, it contains a submodule. Now all these submodules are null spaces w.r.t. the sesquilinear form, i.e. they are orthogonal to any state in V (λ). Therefore one can consistently factor them out, and obtain a sesquilinear form on the quotient space. To see that they are null, let vµ ∈ V (λ) be in some submod-
ule, i.e. wλ ∈ / U · vµ . Now for any v ∈ U · wλ , it follows (vµ , v) ∈ (Uvµ , wλ ) = 0,
using invariance and the fact that there is only one vector with weight λ in the h.w. representation V (λ), namely wλ . Conversely,
Lemma 1.2.2 Let wλ be the highest weight vector of an irreducible highest weight representation L(λ) with invariant inner product. If (wλ , wλ ) 6= 0, then ( , ) is
nondegenerate, i.e.
det(L(λ)η ) 6= 0
(1.84)
for every weight space with weight λ − η in L(λ). Proof
Assume to the contrary that there is a vector vµ which is orthogonal to all
vectors of the same weight, and therefore to all vectors of any weight. Because L(λ) is irreducible, there exists an u ∈ U such that wλ = u · vµ . But then (wλ , wλ) =
(wλ , u · vµ ) = (u · wλ , vµ ) = 0, which is a contradiction. ⊔ ⊓ 19
Verma modules For any weight λ, there exists a ”universal” highest weight module, the Verma module M(λ). It is the (unique) U - module having a h.w. vector wλ
such that the vectors Xk− wλ form a basis of M(λ) [13], where the Xk− are a P.B.W. basis of U − . This is the only infinite–dimensional representation we will consider, and only as a technical tool. The importance of Verma modules lies in the fact that all
highest weight representations can be obtained from M(λ) by factoring out an appropriate submodule. In particular, the (unique) irrep L(λ) with h.w. λ is obtained from M(λ) by factoring out its maximal proper submodule. Since it is a highest weight module, one can define a unique invariant inner product ( , ) on a Verma module for |q| = 1 and q ∈ IR, and its maximal proper submodule is precisely the corresponding
null subspace (see [13] on how to define analogous forms for generic q).
| Forms on tensor products Now let Vi be h.w. representations of U for any q ∈ C
with dominant integral highest weight µi , such that the Vi are irreducible as long as
q is not a root of unity. Therefore on each Vi there is an invariant inner product ( , )i for q ∈ IR and for |q| = 1, which is non–degenerate if q is not a root of unity.
It is important to realize that the representation space Vi is the same for all q (in particular for q = 1), only the action of U on it depends on q, and is in fact analytic
(one way to see this is to use a P.B.W. basis, another is to construct the Vi by taking suitable tensor products, as we will see in a moment). Let V ≡ V1 ⊗ ... ⊗ Vr ,
(1.85)
and for a = a1 ⊗ ... ⊗ ar ∈ V1 ⊗ ... ⊗ Vr and b ∈ V , define (a, b)⊗ ≡ (a1 , b1 )1 ...(ar , br )r .
(1.86)
We claim that for q ∈ IR, ( , )⊗ is a positive–definite inner product:
( , )⊗ is invariant because of (1.70) for q ∈ IR, and it is certainly hermitian and (i)
positive definite if the ( , )i are. Let Mki ,li be the hermitian matrix of ( , )i in (i)
some basis of Vi . Since the ( , )i are determined by the algebra alone, the Mki ,li are certainly continuous (and can be extended to analytic objects), so their eigenvalues are real and continuous. Since Vi remains irreducible for q ∈ IR as shown above and the eigenvalues are positive for q = 1, they cannot vanish for q ∈ IR because of Lemma 1.2.2. So ( , )⊗ is indeed a positive–definite invariant inner product. Similarly, ( , )⊗
| if it is built from bilinear forms on the V . is an invariant bilinear form for any q ∈ C i
Now for q ∈ IR, one can use the Gram–Schmidt orthogonalization method as
usual, and V is the direct sum of orthogonal highest weight irreps Vλl with the same 20
highest weights λl and multiplicities mλl as classically. This implies that for q ∈ IR, the Drinfeld Casimir v satisfies the characteristic equation Y λl
(v − q −cλl ) = 0,
(1.87)
where the product is over all different highest weights counted once, and not with | , and one can write multiplicity mλl . Since v is analytic, (1.87) holds for all q ∈ C
down the projectors on the eigenspaces of v as Pλl = with
P
Q
−cλ ′ l ) λ′l 6=λl (v − q , Q −c −cλl − q λl′ ) λ′l 6=λl (q
(1.88)
Pλl = 1 and Pλl Pλk = δλl ,λk Pλl . They only singularities are isolated poles in q,
and it follows that for generic q, the image of Pλl consists of mλl copies of the highest weight irrep Vλl with h.w. λl . In fact using Lemma 1.2.1, this may break down only at roots of unity. Thus we have shown complete reducibility of V for q not a root of unity. This argument illustrates the use of inner products. From now on, we will only consider the case |q| = 1. Then one can define another invariant sesquilinearform on
V = V1 ⊗ ... ⊗ Vr , namely
(a, b)R ≡ (a, R12...l b)⊗
(1.89)
with R12...l as in Lemma 1.1.1. Indeed using (1.75) and (1.74), (x · a, b)R = (∆(r) (x)(a1 ⊗ . . . ⊗ ar ), R1...r (b1 ⊗ . . . ⊗ br ))⊗ = (a1 ⊗ . . . ⊗ ar , ∆′(r) (x)R1...r (b1 ⊗ . . . ⊗ br ))⊗
= (a1 ⊗ . . . ⊗ ar , R1...r ∆(r) (x)(b1 ⊗ . . . ⊗ br ))⊗ = (a, x · b)R ,
(1.90)
since the ( , )i are invariant w.r.t. x, and ∆′(r) is the flipped r – fold coproduct. While it is not positive definite in general, ( , )R is nondegenerate if q is not a root of unity, (i)
which will be very important later. To see this, let again Mki ,li be the invertible
matrix of the inner products ( , )i on the irreps Vi . Then the matrix of ( , )R is P
k′
(1)
(r)
k ′ ,...,k ′
Mk1 ,k′ . . . Mkr ,kr′ Rl11,...,lrr , which is invertible, because R12...r is invertible. In fact, 1
( , )R remains nondegenerate at roots of unity as long as all the Vi remain irreducible,
since then R12...r exists and is invertible on these representations, as we will see in section 1.2.3.
21
In the classical limit q → 1, ( , )R reduces to ( , )⊗ since R → 1 ⊗ 1, however it is
not hermitian unless q = 1 (remember |q| = 1). In chapter 4, we will show how one can define a (hermitian) inner product on a ”part” of V using a BRST operator for q
a root of unity, and in fact a many–particle Hilbert space, with the ”correct” classical limit. Therefore we have to study the root of unity case. But first, we briefly discuss the ˆ –matrix: R
1.2.2
ˆ Matrix and Centralizer Algebra R–
For a (finite–dimensional) representation V of U, consider the n–fold tensor product
of V with itself,
n
V ⊗ ≡ V ⊗ ... ⊗ V.
(1.91)
This carries a natural representation of U using the n–fold coproduct ∆(n) . Classically,
the symmetric group (or its group algebra) generated by τi,i+1 which interchanges the n
factors in position i and i+1 commutes with the action of U(g) on V ⊗ . The maximal such algebra commuting with the representation is called the centralizer algebra. In the quantum case, there is an analog of this, namely ˆ i,i+1 ≡ id ⊗ ... ⊗(τ ◦ (π ⊗ π)R) ⊗ ... ⊗ id R
(1.92)
where the nontrivial part is in positions i and i + 1, and π is the representation on V . Notice that such a definition only makes sense for identical representations. It follows ˆ i,i+1 commutes with the action of U on V ⊗n . from (1.27) and coassociativity that R Therefore representations of U on this space fall into representations of the centralizer
algebra. This is familiar from quantum field theory, where bosons and fermions are totally symmmetric resp. antisymmetric representations of the permutation group. The Yang–Baxter equation now becomes ˆ i,i+1 R ˆ i+1,i+2 R ˆ i,i+1 = R ˆ i+1,i+2 R ˆ i,i+1 R ˆ i+1,i+2 R
(1.93)
ˆ −2 (v ⊗ v). Now v is diagonalizable Acting on V ⊗ V , (1.53) becomes ∆(v) = (R) ˆ 2 for generic q with eigenvalues q −cλ , because of complete reducibility. Therefore (R) is diagonalizable, with nonzero eigenvalues. This implies (e.g. using the Jordan ˆ is diagonalizable for generic q, with eigenvalues normal form, cp. section 4.1) that R 1
±q 2 (cλ −2cµ ) where µ is the highest weight of V if V is irreducible. Such a result was
first obtained in [49] using a different method. 22
The centralizer algebra provides a connection between quantum groups and conformal field theory [23, 1]. For small representations, it can be described explicitely, and again the root of unity case is very different from the generic case.
1.2.3
Aspects of Representation Theory at Roots of Unity
Generally speaking, the root of unity case is somewhat more complicated than the case of generic q, but also more interesting and probably more relevant to physics. Unfortunately, the general mathematical literature on this subject is not very accessible to physicist6 . The rank one case (i.e. Uq (sl(2)) and its real structures) however is quite instructive and is discussed in [30, 45]. In this section we will only mention a few important features, and many of the later sections will be devoted to studying certain aspects in more detail. In general, it is probably fair to say that the root of unity case is not well enough understood. First, the subalgebras of Uq (g) generated by Xi± and Hi are nothing but Uq (sl(2)) algebras (the coalgebar structure is not the same, however), with qi instead of q. Let q = e2πin/m
(1.94)
with gcd(m, n) = 1, and let M = m if m is odd, and M = m/2 if m is even. Similarly for qi = e2πidi n/m , let M(i) = m if di =
1 , 2
and M(i) = M if di = 1 (recall our
normalization conventions in section 1.1.2). Then M(i) is the smallest integer such that [M(i) ]qi = 0. Highest weight vectors and irreducible representations
(1.95) The following cru-
cial formula can be checked easily [30]: [Xi+ , (Xi− )k ] = (Xi− )k−1 [k]qi [Hi − k + 1]qi .
(1.96)
In particular, this shows that (Xi− )M(i) is central in U, and so is (Xi+ )M(i) . Now if wλ
is a highest weight vector, then (Xi− )M(i) · wλ is either zero or again a highest weight
vector. In the latter case, the representation contains an invariant submodule. In particular, (Xi− )M(i) · wλ = 0 6
[13] is among the more readable sources.
23
(1.97)
on all irreps with highest weight vector wλ . Due to the braid group action (1.46) resp. algebra automorphism, similar statements apply to all root vectors Xβ±r , and considering the P.B.W. basis of U, it follows that all irreducible highest weight rep-
resentations are finite–dimensional at roots of unity. This is very different from the generic case. Another important feature is the existence of non–trivial one–dimensional representations at roots of unity, namely wλ0 with weight λ0 =
P
m kα 2n i i
for integers ki . It
is easy to check from (1.15) that this is indeed a representation of U. By tensoring any
representation with wλ0 , one obtains another representation with identical structure, but all weights shifted by λ0 .
There exist also ”cyclic” representations with (Xi− )M(i) = const if q Hi is used instead of Hi , see [6]. Assume V (λ) is a highest weight module of U which is analytic in q (i.e. the
vector space V (λ) is fixed, but the action of U on it depends analytically on q, such
as a Verma module with dominant integral λ). The submodules contained in V (λ) for generic q will certainly survive at roots of unity, since a highest weight vector w P
is characterized by (
i
Xi+ ) · w = 0, which at roots of unity may have more, but not
fewer solutions than generically. In fact, highest weight modules typically develop
additional h.w. vectors at roots of unity. We can see this in the example of a Verma module of Uq (sl(2)): Let M(j) be the Verma module of Uq (sl(2)) with highest weight λ = j, i.e. H · wj = jwj and X + · wj = 0. Then M(j) has a basis {wj , (X − )k · wj ;
k ∈ IN}
with weights j, j − 2, . . . . For generic q, M(j) contains another highest weight vector
only if j ∈ IN, namely with weight −j − 2; this can be seen from (1.96). However
for q a root of unity, [H − k + 1]q = 0 if H − k + 1 = M, and (1.96) implies that there is an additional h.w. vector at weight j − 2k = j − 2(j + 1 − M) = 2M − j − 2
(if this is smaller than j and j ∈ ZZ), another one at weight j − 2M, and so on.
In fact, the weights of all the h.w. vectors in M(j) can be obtained from j by the
action of the ”affine Weyl group” generated by reflections σl with reflection centers lM − ρ = lM − 1, for any l ∈ ZZ. An analogous statement (the ”strong linkage
principle”) holds in the higher rank case as well, and will be discussed in section 3.2.3. This can be used to determinine the structure of the irreps of U.
In summary, the highest weight irreps at roots of unity are ”usually” smaller than
the irreps with the same highest weight for generic q, and they are always finite– dimensional. 24
Tensor products The coproduct determines the representation of U on a tensor
product, and for q as in (1.94), one can easily see using a q –binomial theorem M(i) Hi /2
that ∆(Xi± )M(i) = (Xi± )M(i) ⊗ qi
−M(i) Hi /2
+ qi
⊗(Xi± )M(i) , cp. [45]. Therefore
if Vi are highest weight irreps, then (Xi± )M(i) = 0 on V1 ⊗ V2 , and similarly for any
number of factors. In this context, it is useful to consider the various quantities as being analytic in h ≡ q ′ − q, where q is fixed to be (1.94). Then e.g. [M(i) ]qi′ has a first–order zero in h. In particular, (Xi± )(M(i) ) ≡
M
(Xi± ) (i) [M(i) ]qi
is well–defined, and the
distinction between the unrestricted and restricted specialization mentioned in section 1.1.2 becomes important. We will essentially work in the unrestricted specialization, i.e (Xi± )(M(i) ) is not considered an element of U.
Consider the tensor product V1 ⊗ V2 of two representations V1 , V2 which are irre-
ducible for generic q. It is well known (e.g. [45, 30]) that if the Vi are ”large enough”, V1 ⊗ V2 does not decompose into the direct sum of irreps at roots of unity, but different
generic irreps (”would–be irreps”) in V1 ⊗ V2 combine into irreducible representations;
this will be discussed in detail in later sections. One should notice that this can happen because 1) all Casimirs, including the Drinfeld Casimir v, approach the same
value on the ”would–be irreps” which recombine as q ′ → q, and 2) the larger of the
recombining ”would–be irreps” develops a h.w. vector, which becomes the h.w. vec-
tor of the smaller constituent. In other words, the image of different projectors (1.88) becomes linearly dependent at roots of unity, and they develop poles. Nevertheless, Lemma 1.2.3 The image Im(Pλ ) of Pλ (1.88) for any given λ is analytic even at roots of unity, in the sense that there exists an analytic basis for it. In particular, the dimension is the same as generically. Proof
One can inductively define an analytic basis of Im(Pλ (q ′ )) for q ′ near the
root of unity q as follows: Suppose the {vi (q ′ )}di=1 are analytic and linearly indepen-
dent at q ′ = q, and satisfy Pλ (q ′ ) · vi (q ′ ) = vi (q ′ ). If d is smaller than the generic
dimension of Im(Pλ ), take a vector vd+1 ∈ Im(Pλ (q0 )) for q0 near q which is not
in the span of the {vi (q ′ )}di=1 at q ′ = q0 . Define vd+1 (q ′ ) ≡ hk Pλ (q ′ ) · vd+1 , where
k ∈ ZZ is such that vd+1 (q ′ ) is analytic and non–vanishing at q ′ = q (this is possible
because Pλ (q ′ ) has only poles). Then vd+1 (q ′ ) satisfies Pλ (q ′ ) · vd+1 (q ′ ) = vd+1 (q ′ ) for q ′ 6= q, since the Pλ are projectors. Furthermore, {vi (q ′ )}d+1 i=1 are linearly independent
except possibly for isolated values of q ′ , and if they are linearly dependent at q ′ = q, one can redefine v˜d+1 (q ′ ) =
1 P ( vd+1 (q ′ ) hk
− ai vi (q ′ )), so that the new {vi }d+1 i=1 span
the same space at q ′ 6= q, are analytic and linearly independent at q ′ = q. This is 25
always possible, because the determinant defined by the {vi (q ′ )}d+1 i=1 is analytic, but not identically zero. ⊔ ⊓
Notice that it is essential that the Pλ have only poles at q ′ = q, and no essential singularities. Furthermore, if a vector w is not in ⊕λ Im(Pλ ) at the root of unity, then it is clear that Pλ (q ′ ) · w will indeed have a pole at q ′ = q for some Pλ .
R at roots of unity Finally we need to know whether R makes sense at roots
of unity. This can be answered using the explicit formulas for R given in [33, 32, ˜ β , i.e. universal R’s of 34], refining (1.47). It turns out that R is built out of R i the Uq (sl(2)) subalgebras corresponding to all roots. Looking at (1.50), the term 1 ((X + )l [l]q !
⊗(X − )l ) becomes ill–defined at roots of unity. So strictly speaking R does
not exist as element of U ⊗ U, but its action on representations V1 ⊗ V2 is well–defined
at roots of unity provided (Xi± )M(i) ·Vi vanishes on all representations Vi . In particular, R is well–defined if all Vi are irreps, and then all the formulas for R hold by analyticity.
The same is true for its many–argument cousin R12...l .
It should be obvious by now that we are dealing with a structure which is very
different from the usual representation theory of Lie groups and algebras. The most remarkable objects however are the indecomposable representations which have barely been mentioned. They will be studied in later sections. But first, we make a digression and consider quantum spaces.
26
Chapter 2 Quantum Spaces associated to Quantum Groups The classical Lie groups SL(N), SO(N) and Sp(N) act naturally on N–dimensional vector spaces M (”vector representation”), preserving certain objects such as volume–
elements or bilinear forms. There exists a perfect analog for quantum groups, introduced by Faddeev, Reshetikhin and Takhtadjan [18]. In the spirit on noncommutative geometry, one does not consider the spaces themselves, but the algebras of functions F un(M) on them, which upon quantization turn into noncommutative algebras F un(Mq ). Because it is customary in the literature, we will use the dual formulation of quantum groups in this chapter, namely F un(Gq ) as explained in section 1.1.3.
2.1 2.1.1
Definitions and Examples Actions and Coactions
So far, with ”representation” we always meant a left action of U on a vector space
V . In this chapter, we will be more explicit, and instead of writing x · v we will write x ⊲ v for v ∈ V . A left action of an algebra A on a vector space V is defined by (xy) ⊲ v = x ⊲ (y ⊲ v),
1⊲v =v
(2.1)
for x ∈ A, and V is called a left U-module. If the representation space is not
only a vector space but also an algebra F and A is a Hopf algebra (such as Uq (g)),
we can in addition ask that this action preserve the algebra structure of F , i.e. 27
x ⊲ (ab) = (x(1) ⊲ a)(x(2) ⊲ b) and x ⊲ 1 = 1ǫ(x) for all a, b ∈ F and x ∈ A. F is then
called a left A-module algebra.
Similarly, a right action of A on a vector space V ′ is defined by v ′ ⊳ (xy) = (v ′ ⊳ x) ⊳ y,
v′ ⊳ q = v′,
(2.2)
and V ′ is called a right A-module; correspondingly one defines right A–module alge-
bras.
Just like the comultiplication is the dual operation to multiplication, right or left coactions are dual to left or right actions, respectively. A left coaction of a coalgebra C (i.e., C is equipped with a coproduct) on a vector space V is defined as a linear map ∆C : V → C ⊗ V :
′
v 7→ ∆C (v) ≡ v (1) ⊗ v (2) ,
(2.3)
such that (id ⊗ ∆C )∆C = (∆ ⊗ id)∆C ,
(ǫ ⊗ id)∆C = id.
(2.4)
The prime on the first factor marks a left coaction. If C is a Hopf algebra coacting on
an algebra F , we say that F is a right C-comodule algebra if ∆C (a · b) = ∆C (a) · ∆C (b) and ∆C (1) = 1 ⊗ 1, for all a, b ∈ F . Similarly one defines right comodule algebras.
Now if the coalgebra C is dual to an algebra A in the sense of (1.32), then a left
coaction of C on V induces a right action of A on V and vice versa, via ′
v ⊳ x ≡< x, v (1) > v (2) ,
(2.5)
and right coactions induce left actions. More on these structures can be found in [41, 51]. For our purpose, we will consider left coactions of F un(Gq ) on left comodule algebras F un(Mq ), which according to the above corresponds to right actions of U = Uq (g) on F un(Mq ). Notice that for quantum groups, a left U–module algebra
F can always be transformed into a right U–module algebra and vice versa using the
(linear!) Cartan–Weyl involution: a⊳ x ≡ θ(x) ⊲ a for a ∈ F and x ∈ U. Alternatively,
one could use the antipode instead of θ, but this is a priori not compatible with the algebra structure of F .
2.1.2
Quantum Spaces and Calculus as Comodule Algebras
First a word on the conventions. We have seen in section 1.1.4 that F un(Gq−1 ) is dual to Uq (g). However most of the literature on quantum spaces uses F un(Gq ), and 28
therefore we will do the same in this section. We may later have to replace q by q −1 when we make contact with Uq (g). Recall that F un(Gq ) is the algebra generated by matrix elements Aij with relations (1.38) ik n i k ˆ nm ˆ mn R Am j Al = An Am Rjl ,
(2.6)
ˆ ik = Rki and Rik is the N–dimensional vector representation of R. This is where R mn mn mn ˆ – matrix commutes with the action of U, in the nothing but the statement that the R
ˆ depends on the group and is given e.g. in [18]. dual picture. The explicit form of R
Unless we are dealing with F un(GLq (N)), this has to be supplemented by additional relations corresponding to invariant bilinear forms or determinants (otherwise U and F un(Gq ) are not dual, cp. [41]).
Quantum Euclidean group and space We only consider the case of F un(SOq (N)) and its real forms in detail1 . In that case, the tensor product of 2 vector representations contains a trivial representation corresponding to the invariant bilinear form. ˆ – matrix, which by virtue of section 1.2.2 decomposes into This can be seen from the R ˆ ij = (qP + − q −1 P − + q 1−N P 0 )ij . The metric gij is then determined 3 projectors [18] R kl kl
by (P 0 )ij kl =
q 2 −1 g ij gkl , (q N −1)(q 2−N +1)
where gik g kj = δij . Explicitely, gij = δi,j ′ q −ρi ,
(2.7)
where j ′ = N + 1 − i and ρi are the values of the Weyl vector ρ˜ in the vector
representation. For SOq (N) with N odd, ρi = (N/2−1, N/2−2, ...1/2, 0, −1/2, ..., 1− N/2). Furthermore, Dji ≡ δi,j q −2ρi = g ik gjk generates the square of the antipode (see
section 1.1.4; notice the replacement q → q −1 pointed out in 1.1.3). The last equality follows from Proposition 4.3.2.
In the language of coactions, invariance of gij becomes gij Aik Ajl = gkl ,
(2.8)
which must be imposed on F un(SOq (N)). In section 4.3, we will find a very interesting interpretation of gij , which will show various consistency conditions between gij ˆ and the R–matrix. They can be used to show that (2.7) is consistent, in particular that the lhs is central in F un(SOq (N)). We refer to [18] or [44] for more details. 1
Here, the series Bn and Dn can be treated simultaneously.
29
Similarly, the tensor product of N vector representations contains a trivial representation corresponding to the totally antisymmetric tensor, Aij11 ....AijNN ǫjq1 ....jN = ǫqi1 ....iN ,
(2.9)
and ǫq also satisfies certain consistency conditions. Both gij and ǫqi1 ....iN depend analytically on q and reduce to the classical expressions as q → 1.
Now (the algebra of functions on) quantum Euclidean space F un(EqN ) [18] is gen-
erated by xi with commutation relations k l (P − )ij kl x x = 0.
(2.10)
The center is generated by 1 and r 2 = gij xi xj . One can go further and define algebras of differential forms, derivatives, and so on, see [58, 44, 64]. The algebra of differential k l i j forms is defined by (P + )ij kl dx dx = 0 and gij dx dx = 0, i.e.
ˆ ij dxk dxl . dxi dxj = −q R kl
(2.11)
The epsilon–tensor is then determined by the unique top - (N-) form dxi1 ...dxiN = ǫqi1 ...iN dx1 ...dxN ≡ ǫqi1 ...iN dNx.
(2.12)
One can introduce derivatives which satisfy k l (P − )ij kl ∂ ∂ = 0,
(2.13)
ˆ −1 )ij xk ∂ l , ∂ i xj = g ij + q(R kl
(2.14)
and ˆ ij dxk ∂ l , ∂ i dxj = q −1 R kl
ˆ ij dxk xl . xi dxj = q R kl
(2.15)
All this is consistent, and represents one possible choice. For more details, see e.g. [44]. It can be checked that all the above relations are preserved under the coaction of F un(SOq (N)) ∆(xi ) = Aij ⊗ xj ≡ xi(1) ⊗ xi(2) ,
∆(dxi ) = Aij ⊗ dxj etc., in Sweedler - notation. 30
(2.16)
Finally, the quantum sphere SqN −1 is generated by ti = xi /r where r is central, so gij ti tj = 1. So far, we have not specified any reality structure, i.e. all the above spaces are complex. To define real quantum spaces, we have to impose a star–structure on F un(SOq (N)) and F un(EqN ), i.e. an antilinear involution on these algebras. Again, one has to distinguish the cases of q ∈ IR and |q| = 1. In this chapter, we will consider
the Euclidean case, which corresponds to q ∈ IR. Later we will consider the Anti–de Sitter case, for |q| = 1.
So from now on q ∈ IR. Then there is a star–structure Aij = g jmAlm gli
(2.17)
extended as antilinear involution, which corresponds to F un(SOq (N, IR)) or F un(SOq (N, IR))2 . The antipode can then be written as S(Aij ) = Aji .
(2.18)
On quantum Euclidean space, the corresponding involution is xi = xj gji [18], which compatible with the left coaction of F un((S)Oq (N)), i.e. ∆(xi ) = ∆(xi ). Even though the metric (2.7) looks unusual because we are working in a weight basis, this is indeed a Euclidean space. The extension of this involution to the differentials and derivatives is quite complicated [44], but this will not be necessary for our purpose. Since r 2 = r 2 , this also induces an involution on the quantum sphere SqN −1 , which becomes the Euclidean quantum sphere3 . In this chapter, we will often write SOq (N) ≡ F un(SOq (N, IR)) for this real
(”compact”) version of F un(SOq (N)), abusing an earlier convention in the dual picture. Similarly, we will write Oq (N) if the determinant condition (2.9) is not imposed for the sake of generality.
2.2
Integration on Quantum Euclidean Space and Sphere
2
These are C ∗ algebras [48].
3
another C ∗ algebra.
31
2.2.1
Introduction
As a first application of this formalism, we will define invariant integrals of functions or forms over q - deformed Euclidean space and spheres in N dimensions. In the simplest case of the quantum plane, such an integral was first introduced by Wess and Zumino [58]; see also [7]. In the case of quantum Euclidean space, the Gaussian integration method was proposed by a number of authors [19, 31]. However, it is tedious to calculate except in the simplest cases and its properties have never been investigated thoroughly; in particular, we point out that determining the class of integrable functions is a rather subtle issue. In this chapter, we will give a different definition based on spherical integration in N dimensions and investigate its properties in detail [55]. Although this idea has already appeared in the literature [24], it has not been developed very far. We first show that there is a unique invariant integral over the quantum Euclidean sphere, and prove that it is positive definite and satisfies a cyclic property involving the D –matrix of SOq (N). The integral over quantum Euclidean space is then defined by radial integration, both for functions and N forms. One naturally obtains a large class of integrable functions. It turns out not to be determined uniquely by rotation and translation invariance (=Stokes theorem) alone; it is unique after e.g. imposing a general scaling law. It is positive definite as well and thus allows to define a Hilbertspace of square - integrable functions, and satisfies the same cyclic property. The cyclic property also holds for the integral of N and N − 1 –forms over spheres,
which leads to a simple, truly noncommutative proof of Stokes theorem on Euclidean space with and without spherical boundary terms, as well as on the sphere. These proofs only work for q 6= 1, nevertheless they reduce to the classical Stokes theorem for q → 1. This shows the power of noncommutative geometry.
Although only the case of quantum Euclidean space is considered here, the general
approach is clearly applicable to other reality structures as well. In particular, we will later consider the case of quantum Anti–de Sitter space, which is nothing but the quantum sphere Sq4 with a suitable reality structure. As expected, an integral can be obtained from the Euclidean case by analytic continuation. We hope that this will eventually find applications e.g. to define actions for field theories on such noncommutative spaces. The conventions are as in the previous section with q ∈ IR except in some proofs.
32
2.2.2
Integral on the Quantum Sphere SqN −1
We first define a (complex - valued) integral < f (t) >t of a function f (t) over SqN −1 . It should certainly be invariant under Oq (N), which means Aij11 ...Aijnn < tj1 ...tjn >t =< ti1 ...tin >t .
(2.19)
Of course, it has to satisfy gil il+1 < ti1 ...tin >t =< ti1 ...til−1 til+2 ...tin >t
ii
< tj1 ...tjn >t = 0 and (P − )jlljl+1 l+1 (2.20)
We require one more property, namely that I i1 ...in ≡< ti1 ...tin >t is analytic in (q −1),
i.e. its Laurent series in (q − 1) has no negative terms (we can then assume that the classical limit q = 1 is nonzero). These properties in fact determine the spherical
integral uniquely: for n odd, one should define < ti1 ...tin >t = 0, and Proposition 2.2.1 For even n, there exists (up to normalization) one and only one tensor I i1 ...in = I i1 ...in (q) analytic in (q − 1) which is invariant under Oq (N) Aij11 ...Aijnn I j1 ...jn = I i1 ...in
(2.21)
and symmetric, ii
(P − )jll jl+1 I j1 ...jn = 0 l+1
(2.22)
for any l. It can be normalized such that gil il+1 I i1 ...in = I i1 ...il−1il+2 ...in
(2.23)
for any l. I ij ∝ g ij .
An explicit form is e.g. I i1 ...in = λn (∆n/2 xi1 ...xin ), where ∆ = gij ∂ i ∂ j is the
Laplacian (in either of the 2 possible calculi), and λn is analytic in (q − 1). For
q = 1, they reduce to the classical symmetric invariant tensors. Proof
The proof is by induction on n. For n = 2, g ij is in fact the only invariant
symmetric (and analytic) such tensor. ′ Assume the statement is true for n, and suppose In+2 and In+2 satisfy the above
conditions. Using the uniqueness of In , we have (in shorthand - notation) g12 In+2 = f (q − 1)In
′ g12 In+2 = f ′ (q − 1)In
33
(2.24) (2.25)
where the f (q − 1) are analytic, because the left - hand sides are invariant, symmet-
′ ric and analytic. Then Jn+2 = f ′ In+2 − f In+2 is symmetric, analytic, and satisfies
g12 Jn+2 = 0. It remains to show that J = 0. Since J is analytic, we can write J i1 ...in =
∞ X
k=n0
i1 ...in . (q − 1)k J(k)
(2.26)
(q − 1)−n0 J i1 ...in has all the properties of J and has a well-defined, nonzero limit as
q → 1; so we may assume J(0) 6= 0. Now consider invariance,
J i1 ...in = Aij11 ...Aijnn J j1 ...jn .
(2.27)
This equation is valid for all q, and we can take the limit q → 1. Then Aij generate the commutative algebra of functions on the classical Lie group O(N), and J becomes ii
J j1 ...jn = 0 implies that J(0) is J(0) , which is just a classical tensor. Now (P − )jlljl+1 l+1 symmetric for neighboring indices, and therefore it is completely symmetric. With g12 J = 0, this implies that J(0) is totally traceless (classically!). But there exists no totally symmetric traceless invariant tensor for O(N). This proves uniqueness. In particular, I i1 ...in = λn (∆n/2 xi1 ...xin ) obviously satisfies the assumptions of the proposition; it is analytic, because in evaluating the Laplacians, only the metric and ˆ - matrix are involved, which are both analytic. Statement (2.23) now follows the R because x2 is central. ⊔ ⊓ Such invariant tensors have also been considered in [19] (where they are called S), as well as the explicit form involving the Laplacian. Our contribution is a self contained proof of their uniqueness. So < ti1 ...tin >t ≡ I i1 ...in for even n (and 0 for
odd n) defines the unique invariant integral over SqN −1 , which thus coincides with the definition given in [24]. From now on we only consider N ≥ 3 since for N = 1, 2, Euclidean space is
undeformed. The following lemma is the origin of the cyclic properties of invariant
tensors. For quantum groups, the square of the antipode is usually not 1. For (S)Oq (N), it is generated by the D - matrix: S 2 Aij = Dli Alk (D −1)kj where Dli = g ik glk (note that D also defines the quantum trace). Then Lemma 2.2.2 For any invariant tensor J i1 ...in = Aij11 ...Aijnn J j1 ...jn , Dli11 J i2 ...l1 is invariant too: Aij11 ...Aijnn Dlj11 J j2 ...l1 = Dli11 J i2 ...l1 34
(2.28)
Proof
From the above, (2.28) amounts to (S −2 Aij11 )Aij22 ...Aijnn J j2 ...jnj1 = J i2 ...ini1 .
(2.29)
Multiplying with S −1 Aii01 from the left and using S −1 (ab) = (S −1 b)(S −1 a) and (S −1 Aij11 )Aii01 = δji01 , this becomes Aij22 ...Aijnn J j2 ...jni0 = S −1 Aii01 J i2 ...in i1 .
(2.30)
Now multiplying with Ali00 from the right, we get Aij22 ...Aijnn Ali00 J j2 ...jni0 = δil10 J i2 ...ini1 .
(2.31)
But the (lhs) is just J i2 ...inl0 by invariance and thus equal to the (rhs). ⊔ ⊓ We can now show a number of properties of the integral over the sphere: Theorem 2.2.3 < f (t) >t
= < f (t) >t
(2.32)
< f (t)f (t) >t
≥ 0
(2.33)
< f (t)g(t) >t
= < g(t)f (Dt) >t
(2.34)
where (Dt)i = Dji tj . The last statement follows from I i1 ...in = Dji11 I i2 ...inj1 . Proof
(2.35)
For (2.32), we have to show that I jn ...j1 gjn in ...gj1 i1 = I i1 ...in . Using the unique-
ness of I, it is enough to show that I jn ...j1 gjnin ...gj1 i1 is invariant, symmetric and normalized as I. So first,
Aij11 ...Aijnn I kn ...k1 gkn jn ...gk1 j1
= gl1 i1 ...gln in Alknn ...Alk11 I kn ...k1 = Alknn ...Alk11 I kn ...k1 gl1 i1 ...gln in =
I ln ...l1 gln in ...gl1 i1 .
(2.36)
ˆ are real), and Aij1 gk j = gl i Al1 . The We have used that I is real (since g ij and R 1 1 1 1 k1 1 ˆ symmetry condition (2.22) follows from standard compatibility conditions between R ˆ is symmetric. The correct normalization can be seen and g ij , and the fact that R easily using g ij = gij for q - Euclidean space. 35
To show positive definiteness (2.33), we use the observation made by [18] that ti → Aij uj
(2.37)
j with uj = u1 δ1j + uN δN is an embedding SqN −1 → F un(Oq (N)) for u1 uN = (q (N −2)/2 +
k l i j q (2−N )/2 )−1 , since (P − )ij kl u u = 0 and gij u u = 1. In fact, this embedding also
respects the star - structure if one chooses uN = u1 q 1−N/2 and real. Now one can write the integral over SqN −1 in terms of the Haar - measure on the compact quantum group Oq (N, IR) [61, 48]. Namely, i
< ti1 ...tin >t =< Aij11 ...Aijnn >A uj1 ...ujn ≡< Aj >A uj ,
(2.38)
(in short notation) since the Haar - measure A is left (and right) - invariant i
i
k
i
k
< Aj >A = Ak < Aj >A =< Ak >A Aj and analytic, and the normalization condition j
i
is satisfied as well. Then < ti tj >t =< Ak Ar >A uk ur and for f (t) = i
j
i
< f (t)g(t) >t = fi gj < Ak Ar >A uk ur =< fi Ak uk = < f (Au)g(Au) >A .
P
j
fi ti etc.,
gj Ar ur >A (2.39)
This shows that the integral over SqN −1 is positive definite, because the Haar - measure over compact quantum groups is positive definite [61], cp. [11]. Finally we show the cyclic property (2.35). (2.34) then follows immediately. For n = 2, the statement is obvious: g ij = Dki g jk . Again using a shorthand - notation, define J 12...n = D1 I 23...n1.
(2.40)
Using the previous proposition, we only have to show that J is symmetric, invariant, analytic and properly normalized. Analyticity is obvious. The normalization follows immediately by induction, using the property shown in proposition (2.2.1). Invariance of J follows from the above lemma. It remains to show that J is symmetric, and the only nontrivial part of that is (P − )12 J 12...n = 0. Define J˜12...n = (P − )12 J 12...n ,
(2.41)
so J˜ is invariant, antisymmetric and traceless in the first two indices (12), symmetric in the remaining indices (we will say that such a tensor has the ISAT property), and analytic. It is shown below that there is no such J˜ for q = 1 (and N ≥ 3). Then as 36
in proposition (2.2.1), the leading term of the expansion of J˜ in (q − 1) is classical and therefore vanishes, which proves that J˜ = 0 for any q. So from now on q = 1. We show by induction that J˜ = 0. This is true for n = 2: there is no invariant antisymmetric traceless tensor with 2 indices (for N ≥ 3). Now assume the statement is true for n even, and that J˜12...(n+2) has the ISAT property. Define K 12...n = g(n+1),(n+2) J˜12...(n+2) .
(2.42)
K has the ISAT property, so by the induction assumption K = 0.
(2.43)
M 145...(n+2) = g23 J˜12...(n+2) = S14 M 145...(n+2) + A14 M 145...(n+2)
(2.44)
Define
where S and A are the classical symmetrizer and antisymmetrizer. Again by the in-
duction assumption, A14 M 145...(n+2) = 0 (it satisfies the ISAT property). This shows
that M is symmetric in the first two indices (1, 4). Together with the definition of M, this implies that M is totally symmetric. Further, g14 M 145...(n+2) = g14 g23 J˜12...(n+2) = 0 because J˜ is antisymmetric in (1, 2). But then M is totally traceless, and as in proposition (2.2.1) this implies M = 0. Together with (2.43) and the ISAT prop˜ it follows that J˜ is totally traceless. So J˜ corresponds to a certain Young erty of J, tableaux, describing a larger - than - one dimensional irreducible representation of O(N). However, J˜ being invariant means that it is a trivial one - dimensional representation. This is a contradiction and proves J˜ = 0. ⊔ ⊓ Property (2.33) (which is also implied by results in [19], once the uniqueness of the invariant tensors is established) in particular means that one can now define the Hilbertspace of square - integrable functions on SqN −1 . The same will be true for the integral on the entire Quantum Euclidean space. The cyclic property (2.34) is a strong constraint on I i1 ...in and could actually be used to calculate it recursively, besides its obvious interest in its own. An immediate consequence of (2.34) is < f (Dt) >t =< f (t) >t , which also follows from rotation invariance of the integral, because D is essentially the representation of the (exponential of the) Weyl vector of Uq (SO(N)).
37
Notice that although it may not look like, (2.34) is consistent with conjugation: even though the D - matrix is real, we have f (Dt) = f (D −1 t).
(2.45)
To see this, take f (t) = ti ; then the (lhs) becomes D(ti ) = D(tj gji ) = Dkj tk gji =
(2.46)
= Dkj tl glk gji = tl gjl gji = (D −1 )il tl
(2.47)
using the cyclic property of g and Dli = gik glk , which is the (rhs) of the above.
2.2.3
Integral over Quantum Euclidean Space
It is now easy to define an integral over quantum Euclidean space. Since the invariant length r 2 = gij xi xj is central, we can use its square root r as well, and write any function on quantum Euclidean space in the form f (xi ) = f (ti , r). We then define its integral to be < f (x) >x =t (r) · r N −1 >r ,
(2.48)
where < f (t, r) >t (r) is a classical, analytic function in r, and < g(r) >r is some linear functional in r, to be determined by requiring Stokes theorem. It is essential that this radial integral < g(r) >r is really a functional of the analytic continuation of g(r) to a function on the (positive) real line. Only then one obtains a large class of integrable functions, and this concept of integration over the entire space agrees with the classical one. It will turn out that Stokes theorem e.g. in the form < ∂i f (x) >x = 0 holds if and only if the radial integral satisfies the scaling property < g(qr) >r = q −1 < g(r) >r .
(2.49)
This can be shown directly; we will instead give a more elegant proof later. This scaling property is obviously satisfied by an arbitrary superposition of Jackson - sums, < f (r) >r =
Z
q
1
dr0 µ(r0 )
∞ X
f (q n r0 )q n
(2.50)
n=−∞
with arbitrary (positive) ”weight” function µ(r) > 0. The normalization can be fixed 2
such that e.g. < e−r >r gives the classical result. If µ(r) is a delta - function, this is simply a Jackson - sum; for µ(r) = 1, one obtains the classical radial integration < f (r)r
N −1
>r =
Z
q 1
dr0
∞ X
n
n
q (q r0 )
n=−∞
38
N −1
n
f (q r0 ) =
Z
0
∞
drr N −1 f (r).
(2.51)
This is the unique choice of µ(r) for which the scaling property (2.49) holds for any positive real number, not just for powers of q. We define f (xi ) to be integrable (with respect to µ(r)) if the corresponding radial integral in (2.48) is finite. We therefore obtain generally inequivalent integrals for different choices of µ(r), all of which satisfy Stokes theorem. Let us try to compare the above definitions with the Gaussian approach. In that case, one does not resort to a classical integral, and determining the class of integrable functions seems to be rather subtle. The Gaussian integration procedure is based on the observation that the integral of functions of the type (polynomial)·(Gaussian) is uniquely determined by Stokes theorem (and therefore agrees with our definition for any normalized µ(r)); one would then like to extend it to more general functions by a limiting process. Lacking a natural topology on the space of functions (i.e. formal power – series), this limiting process is however quite problematic. One way to see this is because there are actually many different inequivalent integrals labeled by µ(r), such a limiting process can only be unique on the (presumably small) class of functions on which the integral is independent of µ(r). Furthermore even classically, although one can calculate e.g. 2
R
2 1 e−r r 2 +1
by expanding it ”properly” (i.e. using
pointwise or L convergence) in terms of Hermite functions, if one tries to expand it 2
formally e.g. in terms of {r n e−r }, one obtains a divergent sum of integrals. Thus the result may depend on the choice of basis and limiting procedure. It is not clear to
the author how to properly integrate functions other than (polynomial)·(Gaussian) in the Gaussian sense, which would be very desirable, because that approach may be applied to some quantum spaces which do not have a central length element [31]. The properties of the integral over SqN −1 generalize immediately to the Euclidean case, for any positive µ(r): Theorem 2.2.4 < f (x) >x
= < f (x) >x
(2.52)
< f (x)f (x) >x
≥ 0
(2.53)
< f (x)g(x) >x
= < g(x)f (Dx) >x ,
(2.54)
= q −N < f (x) >x
(2.55)
and < f (qx) >x if and only if (2.49) holds.
39
Proof
Immediately from theorem (2.2.3), (2.49) and (2.48), using Dr = r and
µ(r0 ) > 0. ⊔ ⊓ (2.52) and (2.55) were already known for the special case of the Gaussian integral [19]. It was pointed out to me by G. Fiore that in this case, positivity was also shown in [20].
2.2.4
Integration of Forms
It turns out to be very useful to consider not only integrals over functions, but also over forms, just like classically. As was mentionned before, there exists a unique N form dxi1 ...dxiN = ǫqi1 ...iN dNx, and we define Z
x
dNxf (x) =< f (x) >x ,
(2.56)
i.e. we first commute dNx to the left, and then take the integral over the function on the right. Then the two statements of Stokes theorem < ∂i f (x) >x = 0 and
R
x
dωN −1 = 0 are equivalent.
The following observation by Bruno Zumino [65] will be very useful: there is a
one - form ω=
1 1 q2 d(r 2) = q dr = dr 2 (q + 1)r r r
(2.57)
where rdxi = qdxi r, which generates the calculus on quantum Euclidean space by [ω, f ]± = (1 − q)df
(2.58)
for any form f with the appropriate grading. It satisfies dω = ω 2 = 0.
(2.59)
We define the integral of a N - form over the sphere r · SqN −1 with radius r by Z
dNxf (x) = ωr N < f (x) >t = drr N −1 < f (x) >t ,
(2.60)
r·SqN−1
which is a one - form in r, as classically. It satisfies Z
q N dNxf (qx) =
r·SqN−1
Z
qr·SqN−1
40
dNxf (x)
(2.61)
R
where (drf (r))(qr) = qdrf (qr). Now defining written as
Z
N
x
d xf (x) =
The scaling property (2.49), i.e. the radial integrals satisfies Z
r
R
x
Z
r
(
r
Z
drg(r) =< g(r) >r , (2.56) can be
dNxf (x)).
(2.62)
r·SqN−1
drf (qr) = q −1
Z
r
R
x
dNxf (x) holds if and only if
drf (r).
(2.63)
dNxf (qx) = q −N
We can also define the integral of a (N − 1) form αN −1 (x) over the sphere with radius
r:
Z
−1
r·SqN−1
αN −1 = ω (
Z
r·SqN−1
ωαN −1 ).
(2.64)
The ω −1 simply cancels the explicit ω in (2.60), and it reduces to the correct classical limit for q = 1. The epsilon - tensor satisfies the cylic property: Proposition 2.2.5
Proof
ǫqi1 ...iN = (−1)N −1 Djii1 ǫiq2 ...iN j1 .
(2.65)
1 κ12...N = (−1)N −1 D 1 ǫ23...N q
(2.66)
Define
in shorthand - notation again. Lemma (2.2.2) shows that κ is invariant. κ12...N is traceless and (q -) antisymmetric in (23...N). Now g12 κ12...N = 0 because there exists no invariant, totally antisymmetric traceless tensor with (N − 2) indices for q = 1, so
by analyticity there is none for arbitrary q. Similarly from the theory of irreducible representations of SO(N) [59], P + 12 κ12...N = 0 where P + is the q - symmetrizer,
1 = P + + P − + P 0 . Therefore κ12...N is totally antisymmetric and traceless (for neighboring indices), invariant and analytic. But there exists only one such tensor up to normalization (which can be proved similarly), so κ12...N = f (q)ǫ12...N . It remains q to show f (q) = 1. By repeating the above, one gets ǫ12...N = (f (q))N (det D)ǫ12...N q q (here 12...N stands for the numbers 1,2,...,N), and since det D = 1, it follows f (q) = 1 (times a N-th root of unity, which is fixed by the classical limit). ⊔ ⊓ Now consider a k - form αk (x) = dxi1 ....dxik fi1 ...ik (x) and a (N −k) - form βN −k (x).
Then the following cyclic property for the integral over forms holds: 41
Theorem 2.2.6 Z
αk (x)βN −k (x) = (−1)k(N −k)
r·SqN−1
Z
q −k r·SqN−1
βN −k (x)αk (q N Dx)
(2.67)
where αk (q N Dx) = (q N Ddx)i1 ....(q N Ddx)ik fi1 ...ik (q N Dx). In particular, when αk and βN −k are forms on SqN −1 , i.e. they involve only dxi 1r
and ti , then
Z
Z
k(N −k)
αk (t)βN −k (t) = (−1)
βN −k (t)αk (Dt).
(2.68)
βN −k (x)αk (q N Dx)
(2.69)
SqN−1
SqN−1
On Euclidean space, Z
x
k(N −k)
αk (x)βN −k (x) = (−1)
Z
x
if and only if (2.63) holds. Notice that on the sphere, dNxf (t) = f (t)dNx. Proof
We only have to show that Z
r·SqN−1
and
Z
r·SqN−1
Z
f (x)dNxg(x) =
dNxg(x)f (q N Dx)
(2.70)
r·SqN−1
dxi βN −1 (x) = (−1)N −1
Z
q −1 r·SqN−1
βN −1 (x)(q N Ddx)i .
(2.71)
(2.70) follows immediately from (2.34) and xi dNx = dNxq N xi . To see (2.71), we can assume that βN −1 (x) = dxi2 ...dxiN f (x). The commutation ˆ ij dxk xl are equivalent to relations xi dxj = q R kl
f (q −1 x)dxj = R((dxj )(a) ⊗ f(1) )(dxj )(b) (f (x))(2) = (dxj ⊳ R1 )(f (x) ⊳ R2 )
(2.72)
where R = R1 ⊗ R2 is the universal R for SOq (N), using its quasitriangular property ˆ ij . f ⊳ Y =< Y, f(1) > f(2) is the right action induced by the left and R(Aj ⊗ Ai ) = R k
l
kl
coaction (2.16) of an element Y ∈ Uq (SO(N)). Now invariance of the integral implies (dxj ⊳ R1 ) < f (x) ⊳ R2 >t = dxj < f (x) >t ,
42
(2.73)
because R1 ⊗ ǫ(R2 ) = 1. Using this, (2.72), (2.61) and (2.60), the (rhs) of (2.71)
becomes (−1)
Z
N −1
q −1 r·SqN−1
βN −1 (x)q
N
Dji dxj
N −1
= (−1)
Dji
Z
dxi2 ...dxiN f (q −1 x)dxj
r·SqN−1
= (−1)N −1 Dji ǫi2 ...iN j ωr N < f (x) >t = ǫii2 ...iN ωr N < f (x) >t Z
=
dxi βN −1 (x),
(2.74)
r·SqN−1
using (2.65). This shows (2.71), and (2.68) follows immediately. (2.69) then follows from (2.63). ⊔ ⊓ Another way to show (2.71) following an idea of Branislav Jurco [28] is to use Z
(αk ⊳ SY )βN −k =
r·SqN−1
Z
αk (βN −k ⊳ Y )
(2.75)
r·SqN−1
to move the action of R2 in (2.72) to the left picking up R1 SR2 , which generates
the inverse square of the antipode and thus corresponds to the D −1 - matrix. This
approach however cannot show (2.34) or (2.54), because the commutation relations of functions are more complicated. (2.67) shows in particular that the definition (2.64) is natural, i.e. it essentially does not matter on which side one multiplies with ω. Now we immediately obtain Stokes theorem for the integral over quantum Euclidean space, if and only if (2.63) holds. Noticing that ω(q N Dx) = ω(x), (2.69) implies Z
x
1 1−q
dαN −1 (x) =
Z
∝
Zx
=
x
Z
x
[ω, αN −1]±
ωαN −1 − (−1)N −1 αN −1 ω
(−1)N −1 αN −1 ω − (−1)N −1 αN −1 ω = 0
(2.76)
On the sphere, we get as easily Z
SqN−1
dαN −2 (t) ∝
Z
[ω, αN −2 ]±
SqN−1
= ω
−1
Z
SqN−1
ω(ωαN −2 − (−1)N −2 αN −2 ω)) = 0 43
(2.77)
using (2.68) and ω 2 = 0. It is remarkable that these simple proofs only work for q 6= 1, nevertheless the
statements reduce to the classical Stokes theorem for q → 1. This shows the power of the q - deformation technique.
One can actually obtain a version of Stokes theorem with spherical boundary terms. Define
qZl r0
qk r
ωf (r) =
qZl r0
qk r
0
0
l−1 X 1 dr f (r) = (q − 1) f (r0 q n ), r n=k
(2.78)
which reduces to the correct classical limit, because the (rhs) is a Riemann sum. Define
q l r0Z·SqN−1
qk r
(
αN (x) =
N−1 0 ·Sq
qZl r0
qk r
Z
αN (x)),
(2.79)
N−1 0 r·Sq
For l → ∞ and k → −∞, this becomes an integral over Euclidean space as defined
before. Then
q l r0Z·SqN−1
q k r0 ·SqN−1
dαN −1
1 = 1−q
qZl r0
Z
(
q k r0 r·SqN−1
ωαN −1 − (−1)N −1 αN −1 ω)
q l r0 Z 1 Z = ( ωαN −1 − 1−q k N−1
=
Z
q r0
q l r0 ·SqN−1
r·Sq
αN −1 −
Z
q k r0 ·SqN−1
Z
qr·SqN−1
αN −1 .
ωαN −1) (2.80)
In the last line, (2.60), (2.64) and (2.78) was used.
2.3
Quantum Anti–de Sitter Space
Let us first review the classical Anti–de Sitter space (AdS space), which is a 4– dimensional manifold with constant curvature and signature (+, −, −, −). It can be embedded as a hyperboloid into a 5–dimensional flat space
with signature (+, +, −, −, −), by
z02 + z42 − z12 − z22 − z32 = R2 ,
(2.81)
where R will be called the ”radius” of the AdS space. Similarly we will consider the 2–dimensional version, defined by z02 + z22 − z12 = R2 (of course there is also a 44
3–dimensional case). The symmetry group (isometry group) of this space is SO(2, 3) resp. SO(2, 1), which plays the role of the Poincar´e group. In fact, the Poincare group can be obtained from SO(2, 3) by a contraction, see e.g. [36]. This space has some rather peculiar features: First, its time–like geodesics are finite and closed. In particular, time ”translations” are a U(1) subgroup of SO(2, 3). The space–like geodesics are unbounded. Furthermore the causal structure is somewhat complicated, but we will not worry about these issues here. With the goal in mind to eventually formulate a quantum field theory on a quantized version of ”some” Minkowski–type spacetime, there are several reason why we choose to work with this space and not e.g. with de Sitter space (corresponding to SO(1, 4)) or flat Minkowski space. First, SO(2, 3) has unitary positive–energy representations corresponding to all elementary particles, as opposed to SO(1, 4) [22], and it allows supersymmetric extensions [63]. Second, the seemingly simpler case of flat Minkowski space is actually mathematically more difficult, because the classical Poincare group is not semi–simple, and the theory of quantum Poincar´e groups is not as well developed as in the case of semi–simple quantum groups. But the main justification comes a posteriori, namely from the existence of finite–dimensional unitary representations of SOq (2, 3) for any spin at roots of unity, and some very encouraging results towards a formulation of gauge theories (=theories of massless particles, strictly speaking) in this framework, which will be presented below. The heavy emphasis on group theory seems justified as a powerful guideline through the vast area of noncommutative geometry.
2.3.1
Definition and Basic Properties
Quantum Anti–de Sitter space (q-AdS space) will be defined as a real form of the complex quantum sphere Sq4 defined above, with an (co)action of SOq (2, 3) which is a real form of F un(SOq (5)) resp. Uq (so(5)). Therefore the algebra of the coordinates ti ≡ xi /r is k l (P − )ij = 0, kl t t
(2.82)
t · t ≡ gkl tk tl = 1.
(2.83)
For |q| = 1, consider the reality structure ti = −(−1)Ei tj gji 45
(2.84)
extended as an antilinear algebra–automorphism. Here Ei = (1, 0, 0, 0, −1) for i =
1, 2, ..., 5 (resp. Ei = (1, 0, −1) in the 2–dimensional case), which will turn out to be
the eigenvalues of energy in the vector representation. It is easy to check that indeed
t · t = t· t. Correspondingly on F un(SOq (2, 3)), one can consider the reality structure Aij = (−1)Ei +Ej g jm Alm gli ,
(2.85)
extended as an antilinear algebra–automorphism. The fact that (..) does not reverse the order is not a problem, since we will not consider the ti as operators, only as ”coordinate functions” which will mainly be used in integrals, e.g. to write down Lagrangians. In fact, in quantum field theory the coordinates are not considered as operators on a Hilbert space. Thus this reality structure on q-AdS space has mainly illustrative character; some reality properties of the integral below however will be used to show hermiticity of interaction Lagrangians (if one would consider the ti as operators on a space of functions on q-AdS space, the adjoint could be calculated from a positive–definite inner product, and would not be given by this reality structure). Observables like energy etc. do exist in our approach, in particular elements in the Cartan subalgebra of Uq (SO(2, 3)) which has a suitable reality structure. This is one of the reasons why we prefer to work with U instead of F un(g). To introduce proper units, define
y i ≡ ti R,
(2.86)
y · y = y iy j gij = R2
(2.87)
for a constant4 R ∈ IR>0 .
So from now on |q| = 1. It is easy to see that (2.84) indeed corresponds to Anti–de
Sitter space for q = 1: consider R2 = y · y = y iy j gij = y 1 y 5 + y 2y 4 + y 3y 3 + y 4y 2 + y 5y 1 , and introduce real variables z i by y1 =
z 0√ +iz 4 , 2
y5 =
z 0√ −iz 4 , 2
y2 = i z
1 +iz 3
√
2
, y4 = i z
1 −iz 3
√
2
,
y 3 = iz 2 . Plugging this into (2.87) gives the classical AdS space. This also shows that Ei = (1, 0, 0, 0, −1) is indeed the energy (in suitable units), and similarly for the 2–dimensional version.
There are other possible reality structures which could define an AdS space for b
b
|q| = 1, such as ti = −ti and Aij = Aij extended as an antilinear involution. This is however not compatible with the identification of the energy in Uq (so(2, 3)) which is acting on it. It will nevertheless be useful in some calculations involving the integral. 4
R is different from r, which has nontrivial commutation relations with forms.
46
Integration. One can define an integral < ti1 ..tin >t on q-AdS space by analytic continuation in q from the integral over the Euclidean sphere Sq4 (This clearly corresponds to the Wick rotation in QFT). It trivially satisfies the same algebraic properties as in the Euclidean case, and is compatible with both reality structures on AdS space: Lemma 2.3.1 For |q| = 1, < ti1 ....tin >t =< ti1 ....tin >t =< tin ...ti1 >t
(2.88)
Define J in ...i1 (q) ≡ I i1 ...in (q −1 ); then for |q| = 1, J in ...i1 = (I i1 ...in )∗ , since q
Proof
is the only complex quantity in the explicit formula in Proposition 2.2.1. Applying b
the above ( ) to the statement of invariance (2.21), one gets Aijnn ...Aij11 J j1 ...jn = J i1 ...in . Now from a slightly generalized Proposition 2.2.1 where (2.21) and (2.22) are required only for |q| = 1, it follows that J in ...i1 (q) = I i1 ...in (q), since J and I are analytic in q. Alternatively, one can consider the anti–algebra automorphism ρ(Aij ) = Aij ,
ρ(q) =
q −1 , where q is treated as a formal variable. Now (2.88) follows from (2.32). ⊔ ⊓ At first sight, it may not look sensible to define an integral of polynomials on a noncompact space. However we are really interested in the case of roots of unity, where the analog of ”square–integrable functions” are indeed obtained as (quotients of ) polynomials, as explained in the following sections. The normalization has to be refined somewhat at roots of unity, and at this point, we make no statement on positivity.
2.3.2
Commutation Relations and Length Scale
Let us write down the algebra of coordinate functions on q-AdS space explicitely. This can be obtained from the Euclidean case [18]. In the 2–dimensional case one finds qy 3 y 1 − q −1 y 1 y 3 = (q 1/2 − q −1/2 )R2 , where y 2 is eliminated by the constraint y · y = R2 . In 4 dimensions we find y i y i+k = qy i+k y i if k > 0 and 2i + k 6= 6, 1/2 −1/2 q −q R2 qy 5 y 1 − q −1 y 1 y 5 = q − 1 + q −1 47
(2.89)
q 1/2 − q −1/2 2 R q − 1 + q −1 1/2 − q −1/2 2 −2 1 5 −1 q R = (q − 1)y y + q q − 1 + q −1
qy 4 y 2 − q −1 y 2 y 4 = (1 − q 2 )y 5y 1 + q
(2.90)
where y 3 is eliminated. The important point here is that these relations are inhomogeneous, and therefore contain an intrinsic length scale L0 ≡
q
|q 1/2 − q −1/2 |R;
(2.91)
notice that (q − 1 + q −1 ) ≈ 1, having in mind that q should be very close to 1. Since
|q| = 1, q = e2πih with h a small number (in fact h = m1 , as we will see below). Then √ L0 ≈ 2πhR. Also, notice that L0 is much bigger that |q − q −1 |R which one might
have expected naively (and which will show up later). In flat Euclidean quantum space for example, the commutation relations are homogeneous, and no length scale appears. To make these commutation relations more transparent, one can approximate
them by [y 5 , y 1] = iL20 and [y 4, y 2 ] = iL20 . As in Quantum Mechanics, this means that the geometry is classical for scales ≫ L0 , and non–classical for scales < L0 . Strictly
speaking, this is only heuristic since the reality structure on the coordinates is not a standard star structure. However it is clear that there really is such a scale, and in the compact (Euclidean) version, the argument is indeed rigorous. This very satisfactory, and the way it should be if this is to find applications in high energy physics. Being extremely optimistic, one is tempted to identify L0 with the Planck scale, where one expects the classical behaviour of space–time to break down. Of course, there is no justification for this so far. It means that q has to be very close to one. These considerations are continued in section 3.2.5.
48
Chapter 3 The Anti–de Sitter Group and its Unitary Representations 3.1
The Classical Case
3.1.1
SO(2, 3) and SO(2, 1)
| )) The classical AdS group is SO(2, 3) resp. U(so(2, 3)), which is a real form of U(so(5, C
and plays the role of the Poincar´e group. The Cartan matrix for its rank 2 Lie algebra B2 is
2 −2 Aij = , −1 2
2 −1 (αi , αj ) = , −1 1
(3.1)
so d1 = 1, d2 = 1/2, to have the standard physics normalization. The weight diagrams of the vector representation V5 and the spinor representation V4 are given in figure 3.1 for illustration; the adjoint V10 is 10 –dimensional. The Weyl vector is ρ = 3 α 2 1
+ 2α2 .
1 2
P
According to (1.22), we define h1 = H1 , h2 = 21 H2 , e±1 = X1± , and e±2 =
α>0
α=
q
[ 12 ]X2± .
Now one can obtain a Cartan–Weyl basis corresponding to all the roots, as explained in section 1.1.4. We choose a slightly different labeling here in order to have (essentially) the same conventions as in [36]. Using the longest element of the Weyl group ω = τ1 τ2 τ1 τ2 , define β1 = α1 , β2 = σ1 σ2 σ1 α2 = α2 , β3 = σ1 a2 = α1 + α2 , β4 = σ1 σ2 α1 = α1 + 2α2 , see figure 3.1.
The corresponding Cartan–
Weyl basis of root vectors is e3 = [e2 , e1 ], e−3 = [e−1 , e−2 ], h3 = h1 + h2 and
e4 = [e2 , e3 ], e−4 = [e−3 , e−2 ], and h4 = h1 + 2h2 . Alternatively one can use the 49
H3
. . . . . . . . .
. . . . . . . . .
1
H2
2
Figure 3.1: Vector and spinor representations of SO(2, 3) braid–group action (1.46) which of course works for the classical case as well, i.e. {e±1 , e±3 , e±4 , e±2 } = {e±1 , T1 e±2 , T1 T2 e±1 , T1 T2 T1 e±2 }. Up to signs, this agrees with
the basis used in [36].
The reality structure and the identification of the usual generators of the Poincare group in the limit R → ∞ can be obtained by considering the algebra of generators leaving the metric invariant, see e.g. [22, 36]. It turns out that the following reality structure corresponds to SO(2, 3): Hi = Hi ,
X1+ = −X1− ,
X2+ = X2− ,
(3.2)
for i = 1, 2. Then e1 = −e−1 ,
e2 = e−2 ,
e3 = −e−3 ,
e4 = −e−4 .
(3.3)
We identify the weights of the vector representation (y 1, y 2 , y 3, y 4, y 5) to be (β3 , β2 , 0, −β2 , −β3 ), see figure 3.1. Then {e±2 , h2 } is a compact SU(2) subalgebra
which acts only on the spacial variables z 1 , z 2 , z 3 in AdS space (2.81). It corresponds to spatial rotations, and we will sometimes write Jz ≡ h2
(3.4)
to indicate that it can be interpreted as a component of angular momentum. Furthermore {e±3 , h3 } is a noncompact SO(2, 1) subalgebra acting on z 0 , z 2 and z 4 . This
is nothing but a 2 dimensional AdS group, and E ≡ h3 50
(3.5)
is the energy since it generates rotations in the z 0 , z 4 –plane. Then Ei = (1, 0, 0, 0, −1) as in (2.85), and ρi is as given in section 2.1.2 for the Euclidean case. The reality
structure (3.2) on U(so(2, 3)) can now be written as x ≡ (−1)E xc (−1)E ,
(3.6)
where xc was defined in (1.73) for x ∈ U(so(2, 3)).
I want to give a brief explanation for the reality structure of SO(2, 1). Let a b+c X = , and L ∈ SL(2, IR). Then det(X) = −a2 − b2 + c2 is the b − c −a quadratic form on 2–dimensional AdS space, which is invariant under X→ L−1 XL. 1 0 , Therefore SL(2, IR) = SO(2, 1), at least locally. Now for K1 ≡ 21 0 −1
0 1 0 −i , K3 ≡ 21 and K± = K1 ± iK2 , then {K± , K3 } is a K2 ≡ 21 1 0 i 0 su(2) Lie algebra. Furthermore Ka ≡ iK1 , Kb ≡ iK2 are purely imaginary, and there-
fore L = exp(i(αa Ka + αb Kb + α3 K3 )) ∈ SL(2, IR) for real parameters αa,b,3 . Now in
a unitary representation of SL(2, IR) (which will be infinite–dimensional), L† = L−1 , † so Ka,b,3 = Ka,b,3 . But this means that K±† = −K∓ , and K3† = K3 .
3.1.2
Unitary Representations, Massless Particles and BRST from a Group Theoretic Point of View
Let us briefly discuss the classical irreducible unitary representations of SO(2, 3) corresponding to elementary particles. Of course, they are all infinite–dimensional. The most important unitary positive–energy irreducible representations are lowest– weight representations V(λ) with lowest weight λ = E0 β3 − sβ2 ≡ (E0 , s) for any E0
and s such that E0 ≥ s + 1, and both integer or both half integer (i.e. λ is integral,
see section 1.2)) [22]. Unitarity will in fact follow from the quantum case. Then s is the spin of an elementary particle with rest energy E0 . For example, a ”scalar field” has s = 0 and E0 ≥ 1, see figure 3.2; it can be realized in the space of functions f (y) on AdS space.
These representations have only discrete weights, nevertheless they become the usual irreps of the Poincare group in the limit R → ∞, with appropriate rescaling.
There also exist remarkable unitary irreps with non–integral weights and all mul-
tiplicities equal to one, namely the so–called Dirac singletons ”Di” for λ = (1, 1/2) and ”Rac” for λ = (1/2, 0) [14]. While it is not clear if they could be of importance 51
E
E=h 3
Jz=h2
Jz b)
a)
Figure 3.2: a) Scalar field and b) Spinor field. The vertical axis is energy, and the horizontal axis is a component of angular momentum.
in a theory of elementary particles, we will pursue them nevertheless. (We will also encounter some more representations with non–integral weights in the root of unity case. For a more general (classical) discussion, see [22].) The massless case should be defined as E0 = s + 1, cp. [22]. In this case, the rest energy E0 is the smallest possible for a unitary representation of given spin. For s ≥ 1, it is qualitatively different from the massive case with the same s: the lowest–weight representations, which are irreducible in the massive cases, develop an invariant subspace of ”pure gauge” states with lowest weight (E0 + 1, s − 1). The
representations however do not split into the direct sum of ”pure gauges” plus the rest, i.e. they are not completely reducible. This means that there is no complete covariant gauge fixing, and to get rid of them and obtain a unitarizable, irreducible representation as required in a quantum theory, one has to factor them out. They
are always null as we will see. This corresponds precisely to the classical phenomenon in gauge theories, which ensures that the massless photon, graviton etc. have only their appropriate number of degrees of freedom. In general, the concept of mass in Anti–de Sitter space is not as clear as in flat space. Also notice that while ”at rest” there are actually still 2s + 1 states, the representation is nevertheless reduced by one irrep of spin s − 1. 52
E
E
Jz
Jz a)
b)
Figure 3.3: a) Photon and b) Graviton, with their ”pure gauge” subspaces.
The massless representations for spin 1 (”vector field”, ”photon”) and spin 2 (”graviton”) are shown in figure 3.3, with their pure gauge subspaces. There are arrows (indicating the group action) into the subspace, but not out of it. To understand the connection with the usual formalism, let us consider the spin 1 case in more detail. Spin one particles are usually described by one–forms, i.e. A(y) = P
P
Ai (y)dy i in the natural embedding of AdS space, where ”automatically”
gij y i dy j = 0. From a group–theoretic point of view, it would be more natural
(and it is in fact unavoidable on q–AdS space) to consider unconstrained one–forms A=
P
Ai (x)dxi , i.e. including the ”radial” component, where xi are the coordinates
of the underlying 5–dimensional flat space. Such a general one–form is an element of (⊕E0 V(E0 ,0) ) ⊗ V5 and vice versa, where (⊕E0 V(E0 ,0) ) is a space of functions on AdS
space spanned by the (unitary) scalar fields V(E0 ,0) , and V5 is the 5–dimensional vector representation. It is easy to see [22] that as representations, V(E0 ,0) ⊗ V5 = V(E0 ,1) ⊕ V(E0 +1,0) ⊕ V(E0 −1,0) ,
(3.7)
see figure 3.4 . V(E0 ,1) is a vector field, V(E0 +1,0) is the space of ”radial” one – forms AR (y)dR which is usually not considered in the flat case, and V(E0 −1,0) is what is
usually called ”longitudinal” modes, which can be killed by the constraint d∗A(y) = 0 53
a)
= Spin 1
radial
longitudinal
Q b)
= Photon
Figure 3.4: One–forms from a group theoretic point of view: a) massive and b) massless case, with BRST operator Q
(”Lorentz gauge”) where ∗A(y) is the Hodge dual of A(y) (in 4 dimensions; notice
that d ∗ (V(E0 ,1) ) ≡ 0, since d ∗ A is a scalar). In fact, the V(E0 −1,0) part has to be
discarded, since it would lead to negative norm states upon canonical quantization.
In the massless case E0 = 2, V(E0 ,1) has a null subspace of pure gauges (consisting of fields A(y) = dΛ(y)) which is isomorphic to V(E0 +1,0) , and must be factored out. The essential and nontrivial point in a gauge theory is to show that the ”pure gauge” subspaces do indeed decouple, so that they can consistently be factored out. Generally in QFT, this is best done using a BRST operator Q, which has the following characteristic properties: 1) The space of pure gauges is the image of Q (at ghost number 0) 2) Q commutes with the S - matrix, the action, etc. 3) Q2 = 0 Then the physical Hilbert space can then be defined as the cohomology of Q at ghost number 0, i.e. Hphys = {Q = 0}/Im(Q) |gh#0, 54
(3.8)
where Im(Q) is the image of Q, and 2) guarantees that this is consistent, i.e. SQ(...) = Q(...). In the standard formulation for photons, {Q = 0} also implies the constraint
d ∗ A = 0 (on the Hilbert space at ghost number 0), but this could as well be imposed
by hand.
Now notice that in the AdS case (3.7), the radial components of a one–form and the subspace of pure gauges are isomorphic, and it is tempting to define an intertwiner Q from the former to the latter. On V(E0 ,1) and V(E0 −1,0) in (3.7), define Q to be 0.
Then Q acting on A indeed satisfies all the properties 1) to 3) of a BRST operator, and the radial component of A plays the role of a ghost; it is indeed a scalar, and anticommuting as a one – form. Notice that we have only one ghost as opposed to 2 in the usual formulation, and accordingly {Q = 0} does not constrain the longitudinal modes to vanish (this has
to be imposed in addition). So this Q does not correspond precisely to the standard BRST operator in an abelian gauge theory1 . Nevertheless we will take the point of view that the above properties 1) to 3) are the characteristic ones, and call our Q a BRST operator as well. Actually, we will relax the requirement Q2 = 0 in the most general setting (see theorem 4.1.2), but it will hold on the sectors of representations relevant to elementary particles. Thus a BRST operator provides a way to define theories of massless elementary particles, i.e. massless unitary irreps. I consider this to be the essential feature of a (abelian) ”gauge theory”, and not some kind of ”local gauge invariance” which is unphysical anyway. Let us try to see if and how all this works in the q–deformed case.
3.2
The Quantum Anti–de Sitter Group at Roots of Unity
The quantum Anti–de Sitter group is simply SOq (2, 3) ≡ Uq (so(2, 3)) as explained
in section 1.1 for |q| = 1, with the same reality structure (3.2) as in the undeformed
case, i.e.
x = (−1)−E xc (−1)E .
(3.9)
This is consistent with all the properties (1.73) to (1.78), as explained in section 1.1.5. We do not consider q ∈ IR, because we will mainly be interested in the roots of unity 1
I wish to thank B. Morariu for discussions on this
55
case. The root vectors in the quantum case are defined by the braid group action as in section 3.1.1, now using the formulas (1.46). We obtain e3 = q −1 e2 e1 − e1 e2 , e4 = e2 e3 − e3 e2 ,
e−3 = qe−1 e−2 − e−2 e−1 ,
e−4 = e−3 e−2 − e−2 e−3 ,
where h1 = H1 , h2 = 12 H2 , e±1 = X1± and e±2 =
h3 = h1 + h2
h4 = h1 + 2h2 ,
(3.10)
q
[ 12 ]X2± . Up to a trivial automor-
phism, (3.10) agrees with the basis used in [36]. The reality structure is e1 = −e−1 ,
e2 = e−2 ,
e3 = −e−3 ,
e4 = −e−4 .
(3.11)
So {e±2 , h2 } is a SUq 21 (2) algebra (but not coalgebra), and the other three {e±β , hβ }
are noncompact SOq˜(2, 1) algebras.
Now we can study q –deformed positive energy representations such as vector fields. As pointed out before, the representation theory is completely analogous to the classical case if q is not a root of unity, at least for finite–dimensional representations. In our case as well, it is easy to see that V(E0 ,0) ⊗ V5 = V(E0 ,1) ⊕ V(E0 +1,0) ⊕ V(E0 −1,0)
(3.12)
as before for E0 ≥ 2, and the representation spaces are the same as classically. Then
everything is as in section 3.1.2, however we will see below that none of these representations is unitary unless q is a root of unity. In the following sections, we will show that for suitable roots of unity, there are unitary representations of SOq (2, 3) corresponding to all the classical ones mentioned
above [56]. They are all finite–dimensional, and obtained from ”compact” representations by a simple shift in energy. Moreover a BRST operator Q will arise naturally, for any spin. We start with the 2–dimensional case, which is technically simpler.
3.2.1
Unitary Representations of SOq (2, 1)
In this section, we will use some results of [30] on SUq (2), where 2J equals H in our notation. SOq (2, 1) is defined by [H, X ± ] = ±2X ± ,
[X + , X − ] = [H]q
(3.13)
∆(H) = H ⊗ 1 + 1 ⊗ H,
∆(X ± ) = X ± ⊗ q H/2 + q −H/2 ⊗ X ± , S(X + ) = −qX + ,
ǫ(X ± ) = ǫ(H) = 0
S(X − ) = −q −1 X − ,
56
S(H) = −H
with the reality structure H = H,
X + = −X −
(3.14)
as explained in section 3.1.1. Comparing with (1.15), this corresponds to the normalization d = (α, α)/2 = 1, but one can easily change to other normalizations by rescaling q, as in section 1.1.2. The irreps of Uq (su(2)) at roots of unity are well – known [30], and we list some facts. As in section 1.2.3, for q = e2πin/m
(3.15)
with positive relatively prime integers m, n let M = m if m is odd, and M = m/2 if m is even. As explained in general, we can assume that (X ± )M = 0
(3.16)
on all irreps (this excludes cyclic representations). Then all finite - dimensional irreps are highest weight (h.w.) representations with dimension d ≤ M. There are two types of irreps:
m z, j = (d−1)+ 2n
• Vd,z = {ejm ;
m m = j, j −2, ..., −(d−1)+ 2n z} with dimension
d, for any 1 ≤ d ≤ M and z ∈ ZZ, where Hejm = mejm
• Iz1 with dimension M and h.w. (M − 1) + M − 1}.
m z, 2n
| \ {Z for z ∈ C Z+
2n r, 1 m
≤r≤
Note that in the second type, z ∈ ZZ is allowed, in which case we will write VM,z ≡ Iz1
for convenience. We will concentrate on the Vd,z – representations from now on.
Furthermore, the fusion rules at roots of unity state that Vd,z ⊗ Vd′ ,z ′ decomposes into ⊕d′′ Vd′′ ,z+z ′
L
p p Iz+z ′
where Izp are the well - known reducible, but indecomposable
representations of dimension 2M, see figure 3.5 and [30]. Let us consider the invariant inner product (u, v) for u, v ∈ Vd,z , as defined in
section 1.2.1, i.e. x is the adjoint of x ∈ U. If ( , ) is positive–definite, we have a
unitary representation.
Proposition 3.2.1 The representations Vd,z are unitarizable w.r.t SOq (2, 1) if and only if (−1)z+1 sin(2πnk/m) sin(2πn(d − k)/m) > 0 for all k = 1, ..., (d − 1). 57
(3.17)
For d − 1
of its lowest–weight submodules, and let Qµ,µ′ be the quotient of it after factoring out all proper submodules of the Uuλ′ .
Let {uλ′′ } be a basis of lowest weight states of Qµ,µ′ . Then Qµ,µ′ = ⊕V(λ′′ ) where
V(λ′′ ) are the corresponding (physical) irreducible lowest weight modules, i.e. Qµ,µ′ is completely reducible. Therefore the following definition makes sense: Definition 3.2.10 In the above situation, let {uλ′′ } be a basis of physical lowest–
weight states of Qµ,µ′ , and V(λ′′ ) be the corresponding physical lowest weight irreps. Then define ˜ (µ′ ) ≡ V(µ) ⊗V Notice that if
m 2n
M
V(λ′′ )
(3.44)
λ′′
is not integer, then the physical states have non–integral weights, and
the full tensor product of two physical irreps V(µ) ⊗ V(µ′ ) does not contain any physical ˜ (µ′ ) is zero, and there seems to be no reasonnable lowest weights. Therefore V(µ) ⊗V
way to get around this.
Again as in section 3.2.2, one might also include a second ”band” of high–energy states. Now 71
˜ is associative, Theorem 3.2.11 If all weights µ, µ′, . . . involved are integral, then ⊗ ˜ (µ′ ) is unitarizable w.r.t. SOq (2, 3). and V(µ) ⊗V Proof
First, notice that the λ′′ are all integral and none of them gives rise to a
massless representation or a singleton. So by the strong linkage principle, none of the Uuλ′ can contain a physical lowest–weight submodule. Also, lowest weight states for generic q cannot disappear at roots of unity. Therefore Qµ,µ′ contains all the physical
lowest–weight states of the full tensor product. Furthermore, no physical lowest– weight states are contained in (discarded states)⊗ (any states). Then associativity follows from associativity of the full ⊗, and the stucture is the same as classically for
energies h3 ≤
m 4n
(observe that ⊗ contains no massless representations, so classically
inequivalent physical representations cannot recombine into indecomposable ones). ⊔ ⊓ In particular, none of the low–energy states have been discarded. Therefore our
definition is physically sensible, and the case of q = e2πi/m with m even appears to be most interesting physically. The highest (”cosmological”) energies available in this ”low–energy band” are of order Emax =
1 hR
in appropriate units, where R is the radius of AdS space, and
h = 1/m. This is by a factor
√1 h
larger than the energy scale L0 ≈
√1 hR
where the
geometry becomes noncommutative, see section 2.3.2. From a QFT point of view, the latter should be the interesting scale. Thus the hierarchy from the curvature radius R to the geometrical scale L0 is the same as between L0 and 1/Emax . This hierarchy has to be large in order to have a large number of physical states available. In any case, this shows that such a theory (imaginary, so far) could in principle accomodate large systems, with an interesting relation between ”cosmological” scales and a geometrical scale (even though we do not advertise this as a cosmologically interesting model).
3.2.6
Massless Particles, Indecomposable Representations and BRST for SOq (2, 3)
Let n = 1 and m = 2M, i.e. q = eiπ/M . V(E0 ,0) will again be called ”scalar field”, V(E0 ,1) ”vector field”, and so on. V(E0 ,s) has a subspace of pure gauges in the ”massless” case, otherwise they are irreducible. They are unitarizable w.r.t. SOq (2, 3) after factoring out the pure gauge states.
72
a)
X
Q X
b)
Figure 3.8: V(λ) ⊗ V5 for the a) massive and b) massless case. The + in b) means sum as vector spaces, but not as representations.
Consider again vectorfields as one–forms. Instead of studying (3.12) at roots of unity, we can study the tensor product of compact representations, using a shift. In particular for λ = −(E0 − M)β3 , consider V (λ) ⊗ V5 at roots of unity. For E0 > 2,
one can easily see that
V (λ) ⊗ V5 = V (λ + β2 ) ⊕ V (λ − β3 ) ⊕ V (λ + β3 ),
(3.45)
which is the same as for generic q. This follows by analycity from the generic case (this is why we consider the shifted representations), since the representations on the rhs remain irreducible at q = eiπ/M according to section 3.2.4 (notice e.g. that the Drinfeld–casimir is different on these representations). In the (shifted) massless case E0 = 2 however, the generic highest weight representation V (λ+β2) has an invariant subspace as in Theorem 3.2.7. In fact, V (λ+β2 ) and V (λ−β3 ) (the ”radial” part according to section 3.1.2) are combined in one reducible, but indecomposable representation, similar as the representations Izp encountered in section 3.2.1; see figure 3.8 for the noncompact case. This kind of phenomenon at roots of unity is well – known, cp. the general discussion in section 1.2.3. It will be analyzed in general in the next chapter, but we can understand it more directly here. As explained in 1.2.3, R is well–defined for q = eiπ/M if acting on irreps with 73
(Xi± )M(i) = 0 such as the compact V (λ) or V(E0 ,s) . Therefore the invariant sesquilinear form (1.89) on V (λ) ⊗ V5 , (v1 ⊗ w1 , v2 ⊗ w2 )R = (v1 ⊗ w1 , Rv2 ⊗ w2 )⊗
(3.46)
is well–defined and nondegenerate for q = eiπ/M (but not positive definite in general). Now it is easy to see that for q = eiπ/M and λ = (M − 2)β3 , V (λ + β2 ) and
V (λ − β3 ) on the rhs of (3.45) are combined into one indecomposable representation:
Let wλ+β2 resp. wλ−β3 be their h.w. vectors for generic q. We know from Theorem 3.2.7 that for q = eiπ/M , V (λ + β2 ) contains a descendant h.w. vector ϕλ−β3 at weight λ − β3 , which means that ϕλ−β3 is orthogonal to any descendant of wλ+β2 , for any
invariant sesquilinear form (while it could happen that this h.w. vector is zero in a given representation, it can be seen easily that this is not the case here). But since our ( , )R is nondegenerate, there must be some vector χ ∈ / V (λ + β2 ) which is not
orthogonal to ϕλ−β3 . Since wλ+β2 is analytic at q = eiπ/M and wλ−β3 is orthogonal to V (λ + β2 ) for |q| = 1, this is only possible if wλ−β3 → ϕλ−β3 ∈ U − wλ+β2 as q → eiπ/M , i.e. the generically independent h.w. modules V (λ − β3 ) and V (λ + β2 ) become dependent, so that one has to include different states such as χ and its descendants
to span the tensor product at roots of unity. Furthermore, notice that χ cannot be a h.w. vector because (ϕλ−β3 , χ)R 6= 0, so X + χ ∈ V (λ + β2 ), and the structure
is really indecomposable. This is the reason for the appearance of indecomposable representations.
BRST operator. At first, this may look complicated. However, the main point is that this is actually very nice from the BRST point of view: In fact, the BRST operator Q which was defined ”by hand” in the generic case is now an element of the center of U, and maps χ into ϕλ−β3 . This is exactly what one would like in QFT. It
is not so surprising, since X + χ ∈ V (λ + β2 ), so some Q ≈
job. We will show that
P
X − X + should do the
Q ≡ (v 2M − v −2M )
(3.47)
is indeed a BRST operator. One could take higher powers of v as well. To see this, consider first the characteristic equation (1.87) of v in the representation V (λ) ⊗ V5 for generic q: (v − q −cλ+β2 )(v − q −cλ−β3 )(v − q −cλ+β3 ) = 0.
74
(3.48)
For compact representations, cλ = (λ, λ + ρ) ∈ 12 ZZ. As q → eiπ/M , the first two
factors above coincide, and (v 4M − 1)2 contains all the factors in (3.48) separately
(and more). So by continuity it follows that (v 4M − 1)2 = 0, which implies Q2 = 0 for q = eiπ/M
(3.49)
on V (λ) ⊗ V5 , since v is invertible.
It remains to show that Q maps some χ into ϕλ−β3 for λ = (M − 2)β3 . We have
seen that as h ≡ (q − eiπ/M ) → 0, wλ−β3 becomes ϕλ−β3 ∈ U − wλ+β2 , so there is a (fixed) u− ∈ U such that u− wλ+β2 = wλ−β3 + hχ(h), ˜ where χ(h) ˜ ∈ V (λ) ⊗ V5 is analytic in h. Applying Q to that equation for generic q, we get Q(
u− wλ+β2 − wλ−β3 ) = Qχ(h), ˜ h
(3.50)
wich using (1.57) becomes − 4M(cλ+β2 u− wλ+β2 − cλ−β3 wλ−β3 ) + o(h) = −4M(cλ+β2 − cλ−β3 )wλ−β3 + o(h) = Q(χ(h)). ˜
(3.51)
Now wλ−β3 = ϕλ−β3 + o(h), and therefore for h = 0, ϕλ−β3 = Qχ
(3.52)
where χ = c−1 χ(0) ˜ and c = −4M(cλ+β2 − cλ−β3 ) 6= 0 using Lemma 1.2.1. Thus 2) and 3) of the properties of a BRST operator in section 3.1.2 hold for our Q, and 1)
would certainly be satisfied in any covariant theory based on SOq (2, 3). States on which Q does not vanish will be called ”ghosts”. It is obvious that all this generalizes to higher spins, and works at roots of unity only.
75
Chapter 4 Operator Algebra and BRST Structure at Roots of Unity In this chapter, we study the tensor product of irreducible representations for any quantum group at roots of unity using a BRST operator Q, generalizing results of the last section. Q allows us to define in a very simple way a new algebra of representations of U similar as in QFT, which has a ”ghost–free” subalgebra with involution. For
the AdS group, this generalizes the physical many–particle Hilbert space introduced
above, and can be used to define correlators. Finally we give a conjecture on complete reducibility, generalizing the standard truncated tensor product used in CFT [40]. The problem of a symmetrization postulate is also briefly discussed.
4.1
Indecomposable Representations and BRST Operator
Let Vi be compact irreps for generic q, i.e. with with dominant integral h.w. µi , which remain irreducible at the root of unity q0 = e2πn/m .
(4.1)
By a simple shift, this also covers e.g. the unitary representations of SOq (2, 3), but excludes the massless representations for the AdS case (they can be built by tensor products). Then for generic q, V ≡ V1 ⊗ ... ⊗ Vl = ⊕V (λl ) 76
(4.2)
with irreps V (λl ) because it is completely reducible. The space V is the same for any q, and the representation of U depends analytically on h ≡ q − q0 .
(4.3)
The projectors Pλl (1.88) may have a pole at h = 0, but no worse singularity. If q is a phase, consider the involution xc = θ(x∗ ) as in (1.74). Now for q0 = e2πin/m with M = m if m is odd and M = m/2 if m is even, the BRST operator Q is defined as in section 3.2.6, Q ≡ (v dM − v −dM )
(4.4)
where d is an integer depending on the group, such that q02dM cλ = 1 for any integral weight λ; notice that cλ is a rational number. Of course Q can be considered for any q. For |q| = 1, it satisfies c
Q = −Q,
(4.5)
since v c = v −1 , see (1.77). If V is completely reducible at q0 , then Q vanishes at h = 0 by construction, using (1.57). In general, it follows that all the eigenvalues of Q are zero at h = 0. This implies that QN = 0 for large enough N (depending on the representation), however Q will not be zero on the indecomposable representations. It is not clear whether Q2 = 0 in general, but this is not essential. As in the previous section, our essential tools will be the sesquilinear forms (a, b)⊗
and (a, b)R on V defined in section 1.2.1, for a = a1 ⊗ ... ⊗ al ∈ V1 ⊗ ... ⊗ Vl and similarly b ∈ V . Furthermore, define
(a, b)k ≡ (a, Qk b)R
(4.6)
for k ∈ IN, which is invariant but degenerate. This will play an important role in
analyzing the structure of V . If q is a phase, then (4.5) implies Q† = −Q, where
†
(4.7)
is the operator adjoint w.r.t. any of these invariant inner products.
Now define vectorspaces Gk ⊂ V as follows: Gk ≡ {ak ∈ V ;
Qk+1 ak = 0 at h = 0} 77
(4.8)
(again, V does not depend on q, only the representation does). At h = 0, QN = 0 for sufficiently large N because of (1.87). Since Q is a Casimir, the Gk are invariant under the action of U at h = 0, and G0 ⊂ G1 ... ⊂ Gk ⊂ ... ⊂ GN = V for some N.
This is a ”filtration”; it is well known that the tensor product at roots of unity has the structure of a filtration [6]. We can now define the quotients Gk = Gk /Gk−1
(4.9)
which are representations again (for h = 0), and as vectorspace, V ∼ = ⊕k Gk
(4.10)
(but not as representation). The Gk with k > 0 will be called ”ghosts”. Roughly
speaking, Gk essentially contains the ak with Qk ak 6= 0, but Qk+1 ak = 0. From the definition, Q maps Gk into Gk−1 at h = 0, and it is injective (i.e. it never vanishes).
In particular, Qk : Gk → G0 is injective as well, and Q(G0 ) = 0. Therefore at h = 0, (a, b)k = 0 if either a or b ∈ Gk−1 , which means that ( , )k is a well–defined sesquilinear form on Gk or more generally on V /Gk−1 ∼ = ⊕l≥k Gl .
From now we will work with these spaces for q = q0 . Consider v acting on V in its
Jordan normal form, v = D + N where D is a diagonal matrix and N nilpotent, and DN = ND. Since v is invertible, its (generalized) eigenvalues are nonzero, and D is invertible. Then v n = D n + nD n−1N + N 2 (...) is the Jordan normal form of v n , so v is diagonalizable if and only if v n is. In particular for a ∈ G0 , Qa = (v dM − v −dM )a = 0, therefore (v 2dM − 1)a = 0 since v is invertible. This means that a is an eigenvector
of v 2dM , and therefore a is contained in the sum of the proper eigenspaces of v, as we have just seen. Conversely of course, any eigenvector of v will be annihilated by Q, since the eigenvalues and the characteristic equation of v are analytic. Therefore we see again that Q is nilpotent on V . In summary, Lemma 4.1.1 For h = 0, Q : Gk → Gk−1
is injective,
(4.11)
and G0 is the sum of the proper eigenspaces of v. Now one can define Hk = Gk /(QGk+1 ).
(4.12)
generalizing (3.8); we do not require that Q2 = 0. Since QGk+1 is invariant, Hk is a
representation of U as well. An element hk ∈ Hk can be represented by an element of 78
Gk (not uniquely of course), which will still be denoted as hk . Using our sesquilinear forms, one can choose them in a particular way:
0 Gk Theorem 4.1.2 Consider V = V1 ⊗ ... ⊗ Vl as above for h = 0, with V ∼ = ⊕kk=0 as vector space (but not as representation). Then one can choose Hl ⊂ Gl such that
( , )k is nondegenerate on Hk , and
(Hk , Hk+l )k = 0
for l ≥ 1.
(4.13)
Furthermore, Gk = Hk ⊕ QHk+1 ⊕ ... ⊕ Qk0 −k Hk0
(4.14)
as vector space. Therefore Hk ∼ = Hk , and since (Hk , Q(...))k = 0, V /Gk−1 is the direct sum of Hk and its orthogonal complement w.r.t ( , )k , as vectorspace. Notice that this is not trivial, since the ( , )k are not positive definite in general. Furthermore, if Q 6= 0, then the representation is indecomposable. Proof
Let k0 be the maximal k such that Gk 6= 0. Then for any hk0 ∈ Hk0 ≡ Gk0 ,
Qk0 hk 6= 0, and since ( , )0 = ( , )R is nondegenerate, there exists some a such that
(a, Qk0 hk0 )0 = (a, hk0 )k0 6= 0. Furthermore from the definition of ( , )k it follows that
(Hk , Gl )k = 0 if l < k. Therefore ( , )k0 is nondegenerate on Hk0 . This implies as usual that any linear form on Hk0 can be written as (hk0 , .)k0 with a suitable hk0 ∈ Hk0 ,
i.e. there is a isomorphism from the dual space Hk∗0 to Hk0 .
In particular, any a0 ∈ G0 defines a linear form on Hk0 by hk0 → (a0 , hk0 )0 .
Therefore there is a unique element i(a0 ) ∈ Hk0 which satisfies
Now define W0k0 ⊂ G0 by
(a0 − Qk0 −k i(a0 ), Hk0 )k = 0.
(4.15)
W0k0 ≡ (id − Qk0 i)G0 ;
(4.16)
then by definition (W0k0 , Hk0 )0 = 0, and since a0 = (a0 − Qk0 i(a0 )) + Qk0 i(a0 ) ∈
W0k0 ⊕ Qk0 Hk0 for any a0 ∈ G0 , we have
G0 = W0k0 ⊕ Qk0 Hk0
with (W0k0 , Hk0 )0 = 0,
(4.17)
and W0k0 is determined uniquely by this requirement. Now we can define Wkk0 ⊂ Gk for any k to be the space which is mapped into W0k0
by Qk ,
Wkk0 = (Qk )−1 W0k0 79
(4.18)
(as set). For any ak ∈ Gk , decompose Qk ak = w0k0 + Qk0 hk0 accordingly. Then
w0k0 = Qk (ak − Qk0 −k hk0 ), and we have ak = wkk0 + Qk0 −k hk0 for wkk0 ≡ ak − Qk0 −k hk0 . This means that
Gk = Wkk0 ⊕ Qk0 −k Hk0 ,
and
(Wkk0 , Hk0 )k = 0
(4.19)
using the definition of ( , )k . This decomposition is unique since Qk : Gk → G0 is injective.
Now for Gk0 −1 , define
Hk0 −1 ≡ Wkk00−1 .
(4.20)
Then obviously Hk0 −1 ∼ = Hk0 −1 , and (Hk0 −1 , Hk0 )k0 −1 = 0 using (4.19). Thus we have shown (4.14) and (4.13) for k = k0 − 1. Of course, (Hk0 −1 , Q(...))k0 −1 = 0.
Now we can repeat this construction: since (Hk0 −1 , Hk0 )k0 −1 = 0 and Qk0 −1 hk0 −1 6=
0 for any hk0 −1 ∈ Hk0 −1 , it follows that ( , )k0 −1 is nondegenerate on Hk0 −1 . Again, this means that for every a0 ∈ W0k0 there exists some hk0 −1 = i(a0 ) ∈ Hk0 −1 such that (a0 − Qk0 −1 i(a0 ), Hk0 −1 )0 = 0. Define W0k0 −1 ≡ (id − Qk0 −1 i)W0k0 ; then W0k0 = W0k0 −1 ⊕ Qk0 −1 Hk0 −1
(4.21)
and (W0k0 −1 , Hk0 −1 )0 = 0. Therefore G0 = W0k0 −1 ⊕ Qk0 −1 Hk0 −1 ⊕ Qk0 Hk0 ,
(4.22)
and we already know (W0k0 −1 , Hk0 )0 = 0. This implies that for Wkk0 −1 ≡ (Qk )−1 W0k0 −1 , Wkk0 = Wkk0 −1 ⊕ Qk0 −1−k Hk0 −1 and (Wkk0 −1 , Hk0 −1 )k = 0. Finally with −1 Hk0 −2 ≡ Wkk00−2 ,
(4.23)
(4.14) and (4.13) follows. Repeating this argument, we arrive at the decomposition as stated. ⊔ ⊓
4.1.1
A Conjecture on BRST and Complete Reducibility
There are many indications that the following extension of Theorem 4.1.2 holds, in the same context: Conjecture 4.1.3
1) H0 is completely reducible, i.e. it is the direct sum of irre-
ducible representations.
80
2) H0 defines an associative, completely reducible modified tensor product of irreducible representations.
Furthermore, it appears that G0 is the sum of the analytic images of the projectors Pλ at h = 0, see Lemma 1.2.3, and G0 ∩ Q(...) is the subspace where they are linearly
dependent. This could be used to show 2) from 1).
In the context of the AdS group, 2) would generalize the definition of the tensor ˜ in sections 3.2.2 and 3.2.5, and include the ”high–energy” bands as well product ⊗
as irreps which are unitary w.r.t. the ”compact” reality structure. This would also generalize the standard truncated tensor product used in the context of CFT, and ˜ and ⊗. ˆ provide a common framework for ⊗
4.2
An Inner Product on G0
In this section, we will define a hermitian inner product on G0 . To do this, we first √ show how to find an element in U which implements v (or many other functions of v) analytically on G0 for q = q0 .
As already noted, v becomes degenerate on different representations at h = 0, because q −cλ does. For a weight λ, define the λ –group gλ of V to be the set of highest weights in the generic decomposition (4.2) of V for which v is degenerate, cλ
q0 l = q0cλ }.
gλ ≡ {λl ; Then define
Q
gλ′ 6=gλ (v − q −cλ − gλ′ 6=gλ (q
Pg λ ≡ Q
−cλ′
)
q −cλ′ )
(4.24)
,
(4.25)
where the products go over all possible λ –groups once. As opposed to Pλ , this is analytic at q0 . It is not a projector in V , but it is a projector for h = 0 in G0 , since then v is diagonalizable on G0 as pointed out in Lemma 4.1.1; it is simply the projection on the eigenspaces of v corresponding to different λ –groups. Now define
X √ 1 Pg λ q − 2 c λ , v≡
(4.26)
gλ
which is an element of U and analytic at q0 , where λ is some element of gλ . It depends
on the choice of λ ∈ gλ , which simply corresponds to choosing a different branch; pick any of them. It is easy to see that on G0 , √ ( v)2 = v
for h = 0, 81
(4.27)
and its inverse on G0 is given by √
v −1 ≡
X
1
Pg λ q 2 c λ .
(4.28)
gλ
This follows because the Pgλ are projectors on G0 as shown above. Using this, we can finally show the following: Proposition 4.2.1 Define √ (a, b)H ≡ cV (a, v · b)R ,
(4.29)
1
where cV = q 2 (c1 +...+cl) and ci is the quadratic Casimir on the Vi . Then for a, b ∈ G0 , (a, b)H is hermitian at h = 0, i.e. it is an inner product on G0 for q = q0 .
Proof
To see this, notice first that ∆(l) v = (Rl...21R12...l )−1 (v ⊗ ... ⊗ v).
(4.30)
This follows inductively by applying (∆(l−1) ⊗ id) to (1.53): ∆(l) (v) = = = =
(∆(l−1) ⊗ id)R21 (∆(l−1) ⊗ id)R12
′ (R−1 12...(l−1) ⊗ 1)((∆(l−1) ⊗ 1)R21 )R12...l
−1
(∆(l−1) v ⊗ v)
(∆(l−1) ⊗ id)R21 (R−1 12...(l−1) ⊗ 1)R12...l
−1
(∆(l−1) v ⊗ v)
−1
−1 (R−1 12...(l−1) ⊗ 1)(R(l−1)...21 ⊗ 1)Rl...21 R12...l
(∆(l−1) v ⊗ v)
−1
(∆(l−1) v ⊗ v)
= (Rl...21 R12...l )−1 (R(l−1)...21 R12...(l−1) ⊗ 1)(∆(l−1) v ⊗ v) = (Rl...21 R12...l )−1 (v ⊗ ... ⊗ v)
using (1.58), (1.59) and Lemma 1.1.1. Furthermore cV (a, b)∗H =
=
(4.31) √
c
v =
√ ∗ a, R12...l ∆(l) ( v)b ⊗ √ = R12...l ∆(l) ( v)b, a ⊗ √ ′ = ∆(l) ( v)R12...l b, a ⊗ √ −1 −1 = b, Rl..21 ∆(l) ( v )a
⊗ −1 −1 −1 b, R12...l R12...l Rl...21 v
·
√
√ = (b, (v −1 ⊗ .. ⊗ v −1 ) v · a)R
= cV (b, a)H 82
√
v −1 . Then
v·a
⊗
(4.32)
for h = 0, using (1.61). ⊔ ⊓ The above definitions may seem a bit complicated at first. We will show below that all this can be formulated in a very simple way, using an extension of U by a
universal element which implements a Weyl reflection. But before that, we show how the above inner product defines a Hilbert space of physical many–body representations
of the Anti–de Sitter group, with the correct classical limit. Then the adjoint of an operator acting on any component of a tensor product is determined by the positive definite inner product, and is guaranteed to have the correct classical limit, since the inner product has (for q 6= 1, its adjoint will act on the entire tensor product, see below). This finally settles any doubts whether the reality structure (1.74) is suitable to describe physical many–body states.
4.2.1
Hilbert Space for the Quantum Anti–de Sitter group
It follows from Theorem 4.1.2 that (a, Q(...))H = 0 for a ∈ G0 , i.e. the image of Q is null with respect to this inner product. Therefore ( , )H induces an invariant inner
product on H0 as defined in section 4.1. Furthermore, ( , )H is nondegenerate on H0 . We want to show that it induces in fact a positive definite inner product on the physical representations ˜ ⊗V ˜ (µl ) = V(µ1 ) ⊗...
M
V(λk )
(4.33)
λk
defined in section 3.2.5, where all V(µl ) are massive (this is not an essential restriction). The terms on the rhs are (quotients of) analytic, generic representations and therefore contained in (a quotient of) G0 . Thus the results of the last section apply, and the eigenvalues of ( , )H on G0 are either positive or negative. It is clear that for low energies, they will be positive as in the classical limit. We claim that they are positive on all the V(λk ) above, and outline a proof: Consider the tensor product V of the infinite–dimensional (generic) lowest–weight modules corresponding to the massive representations in (4.33). For fixed λk , let Gλk be the sum of the lowest–weight submodules of V with lowest weight λk . According to the strong linkage principle, none of them will contain physical lowest-weight vectors, for any q on the arc from 1 to e2iπ/m . This implies that the physical lowest–weight vectors of Gλk are linearly independent of the other Gλ′k for q on the arc from 1 to q = e2iπ/m , and analytic in q. Now we can define ( , )H as in (4.29) for states with weights λk corresponding 83
to physical Gλk , where one has to use the value of the classical quadratic Casimir on Gλk . R is well–defined for such states, as can be seen from (1.47) and the discussion
in section 1.2.3. Therefore ( , )H is hermitian and analytic for q on the arc from 1 to e2iπ/m . For q = e2iπ/m , it reduces precisely to our inner product on G0 . Furthermore, all states with such weights are contained in ⊕Gλk .
Now ( , )H cannot become null on the physical lowest–weight states of Gλk , where
it is non—degenerate for q on the arc from 1 to e2iπ/m , since the Gλk remain linearly √ independent, so v is analytic and invertible, and the other Gλk′ are orthogonal. Therefore the eigenvalues of ( , )H on Gλk are positive, as classically.
4.3
The Universal Weyl Element w
The proper mathematical tool to obtain this inner product and an involution is an element of an extension of U by generators of the braid group ωi, introduced in [33]
and [34]. The wi act on representations of U and implement the braid group action
(1.46) on U via Ti (x) = wi xwi−1 for x ∈ U. All we need is the generator corresponding
to the longest element of the Weyl group, w. Acting on a highest weight irrep, w
maps the highest weight vector into the lowest weight vector of the contragredient representation. It has the following important properties [33, 34]: ∆(w) = R−1 w ⊗ w = w ⊗ wR−1 21 w 2 = vε
(4.34) (4.35)
where ε is a Casimir with ∆(ε) = ε ⊗ ε
(4.36)
(for SLq (2) and SOq (2, 3), ε is +1 for integer ”spin” and −1 for half–integer ”spin”,
see Appendix B). (4.34) justifies the name ”universal Weyl element”. One can also find the antipode and counit of w. Furthermore, for the ”real” quantum groups (=those having only self–dual representations, i.e. all except An , Dn and E6 ), such | ) and its real forms, the following holds [49]: as SOq (5, C
wxw −1 = θSx = S −1 θx.
(4.37)
We will only consider this ”real” case. For ”complex” groups, this gets corrected by an automorphism of the Dynkin diagram. Actually, (4.37) and (4.35) have only been proved explicitely for the SLq (2) case in the literature, therefore we will supply proofs in Appendix B. 84
Similarly, we define w˜ ≡ (−1)E q 2˜ρ w
(4.38)
e −1 x = S θx. e wx ˜ w˜ −1 = θS
(4.39)
with the same properties except
e ≡ (−1)E θ(x)(−1)E implements the reality structure on U according to section Here θx
1.1.5: in the case of SOq (2, 3), E is the energy operator, while the compact case corresponds to E = 0. Denote the (left) action of x ∈ U on a representation V by x ⊲ vi = vj πij (x),
(4.40)
for a basis vi of V . We are mainly interested in (tensor products of) unitary representations. The following will be very useful: Lemma 4.3.1 If an irrep V (λ) is analytic and unitary at a phase q (for the compact involution (1.73), say), then it has a basis which is orthogonal and normalized w.r.t. both its symmetric bilinear form and its invariant inner product, i.e. πji (θ(x)) = πij (x)
and
πji (xc ) = πij (x)∗ ≡ (πij (x))∗ ,
(4.41) (4.42)
in that basis. Proof
First, one can check that this holds for the fundamental representations Vf
(i.e. the spinor representation for SOq (5)) [49]. We will show the general statement inductively by taking tensor products with the fundamental representation. Suppose it holds for πji on V (µ), and onsider V (µ) ⊗ Vf = ⊕V (λi ). All the multiplicities are
known to be one in this case (this can be seen e.g. using the Racah–Speiser algorithm). ij (λi ) ij Then the Clebsch–Gordan coefficients Km (q) defined by vm = Km vi ⊗ vj ∈ V (λi )
satisfy [49]
1
ij ls fsl (q), Km (q)Rij = (−1)νi q 2 (cλi −cµ −cf ) K m ij −1 fji (q). Km (q ) = (−1)νi K m
(4.43) (4.44)
fsl (q) is the Clebsch in the reversed tensor product, and (−1)νi is the same as Here K m
classically (and just a convention unless the factors are identical). The first can be ij seen similarly as in section 1.2.2. Furthermore, Km (q) is real for q ∈ IR.
85
Using this, we can write an invariant inner product of 2 vectors in V (λi ) as in (4.29), 1
ij −1 kl (λi ) (λi ) (q )Knst (q)Rst (vi ⊗ vj , vk ⊗ vl )⊗ (vm , vn )H = q − 2 (cλi −cµ −cf ) Km
fji (q)K flk (q)δ i δ j = K m n k l
= δnm ,
(4.45)
choosing an orthogonal basis of V (λi ), which is unitarizable by assumption. But then the invariant bilinear form of these vectors is (bi)
(λi ) (λi ) (vm , vn )⊗
ij kl = Km (q)Km (q)(vi, vk )(bi) (vj , vl )(bi)
= δnm ,
(4.46)
using (vi , vk )(bi) = δki on the components, by the induction assumption. This means (λi ) that the vm are orthogonal not only w.r.t. the sesquilinear inner product, but also
the above bilinear form. ⊔ ⊓ In the case of SUq (2), this can easily be checked explicitely. We will always use this basis from now on. By inserting (−1)E , one gets the same e Moreover, statement for the shifted noncompact representations, with θ replaced by θ.
the result also holds for the irreducible quotients of analytic representations at roots of unity of a given unitarity type, such as the physical many–particle representations defined in section 3.2.2 and 3.2.5. Now we can get a better understanding of w. ˜ First, it intertwines a unitary representation with its contragredient (=dual) representation, defined by x˜⊲ai = aj πji (S −1 x) (remember that we only consider ”real” groups): e −1 x) ⊲ a x ⊲ (w˜ ⊲ ai ) = (xw) ˜ ⊲ ai = w˜ ⊲ (θS i
e −1 x)) = w = w˜ ⊲ (aj πij (θS ˜ ⊲ (aj πji (S −1 x))
= w˜ ⊲ (x˜⊲ai ).
(4.47) (4.48)
Other important properties are as follows. Define 1
gij ≡ πji (w)q ˜ 2 cλ = (−1)f g ij
(4.49)
where cλ is the classical quadratic Casimir of the representation, and (−1)f is the value of ε. The last equality follows from (4.35), where gij g jl = δil . This is nothing but the invariant tensor (again, for ”real” groups): 86
Proposition 4.3.2 πki (x1 )πlj (x2 )g kl = ǫ(x)g ij
(4.50)
gij πki (x1 )πlj (x2 ) = ǫ(x)gkl
(4.51)
gij = q
1 c 2 λ
πij ((−1)E w)
ˆ sk g tv = (R ˆ −1 )kv . gls R ut lu
(4.52) (4.53)
(4.53) and some more similar relations are contained in [49]. Proof
(4.50) follows from πki (x1 )πlj (x2 )πlk (w) ˜ = πlj (x2 )πli (x1 w) ˜ e −1 x )) = πlj (x2 )πni (w)π ˜ ln (θ(S 1
= πlj (x2 )πni (w)π ˜ nl (S −1 x1 )
= πnj (x2 S −1 x1 ))πni (w) ˜ = ǫ(x)δnj πni (w), ˜
(4.54)
and similarly (4.50). (4.52) follows from the uniqueness of the invariant tensor (cp. Lemma 2.2.2), noting that πki (x1 )πlj (x2 )πkl ((−1)E w) = πkj (x2 q −2˜ρ w)π ˜ ki (x1 ) i e = πkj (q −2˜ρ w˜ θSx 2 )πk (x1 )
i e = πtj (q −2˜ρ w)π ˜ kt (θSx 2 )πk (x1 )
= πtj (q −2˜ρ w)π ˜ tk (Sx2 )πki (x1 ) = πtj ((−1)E w)δti ǫ(x)
(4.55)
is indeed invariant, where we used (1.51). This shows (4.52) up to a constant, which is one because πji (w)π ˜ kj (w) ˜ = πki (vε) and πij ((−1)E w)πjk ((−1)E w) = πik (vε),
(4.56)
which is the same. Finally, (4.53) is obtained by taking representations of
⊔ ⊓
e −1 )R = (1 ⊗ θ)R e −1 . (1 ⊗ w)R(1 ˜ ⊗w ˜ −1) = (1 ⊗ θS
87
(4.57)
Furthermore, from (4.34) one immediately obtains (1.60), (1.61), as well as (4.30). For |q| = 1, it is consistent to define w˜ = w˜ −1
(4.58)
and similarly for w. Correspondingly, we have πji (w) ˜ ∗ = πij (w˜ −1 ).
(4.59)
This can be checked explicitely using the known formulas for w in terms of the wi [33] and their action on a representation [25, 39].
4.4
An Algebra of Creation and Anihilation Operators
In this section, we will show how to define an interesting algebra of creation and anihilation operators, with involution. This allows us to work with states of different particle numbers, and to write down correlators as in QFT. It is however not clear at present how identical particles should be defined, so we only consider distinguishible particles. Denote the states of a physical representation V(λ) of SOq (2, 3)1 by ai . We can make V(λ) into a U –module algebra F (a) (see section 2.1.1) by defining either ai aj = 0, or more somewhat more interestingly by
ai aj = gij a2 ,
ai aj ak = 0
(4.60)
where a2 is a (scalar) variable, and gij as in (4.49). We will use this algebra only to to show how to define an involution, and to write the inner product on G0 in a elegant way. Notice that in the noncompact case, gij 6= 0 only if one of the ”factors”
is a positive and one a negative energy representation (an antiparticle wavefunction,
i.e. shifted by −2Mβ3 as explained in section 3.2.4). We are using the same letter
ai , because both positive and negative energy representations can be obtained as quotients of one ”big” self–dual representation, namely the H0 part of a tensor product
as in Theorem 4.1.2, which is the direct sum of a positive an a negative energy representation. This is very nice from the QFT point of view, and one should keep this in mind for the following. 1
This works for other real groups as well.
88
Given many left U –module algebras F (a) , F (b) , ... generated by ai , bi , ..., one can
define a combined (left) U –module algebra F using the braided tensor product [42]. As vector space, this is simply F = F (a) ⊗ F (b) ⊗ ..., with commutation relations ai bj = (R2 ⊲ bj )(R1 ⊲ ai )γ
(4.61)
| ; similarly for more variables. This definition requires an (arbitrary) for some γ ∈ C
”ordering” a > b > ... of the different algebras. It is consistent because of the standard
properties of R. via
Finally on the vector space U ⊗ F , one can define a cross product algebra U × F xai = (x1 ⊲ ai )x2 ,
or equivalently x ⊲ ai = x1 ai Sx2 .
(4.62)
This is an algebra, because U is a Hopf algebra and F is a (left) U –module algebra. To make the connection with QFT more obvious, we define a (”Fock”) vacuum i
which reduces U × F to its ”vacuum–representation” U × F i by f (a, b, ..)xi ≡ ǫ(x)f (a, b, ...)i.
(4.63)
This is a tensor product of representation of U, and xf (a, b, ...)i = x1 ⊲ f (a, b, ...)x2 i = x ⊲ f (a, b, ...)i.
(4.64)
In particular this contains the subspace G0 where v is diagonalizable, and we can apply the results of sections 4.1 and 4.2. Define
with
√
√ Ω = w˜ v −1
(4.65)
v −1 as in (4.28). Acting on G0 , this satisfies Ω2 = ε
(4.66)
at roots of unity, using (4.35) and S(v) = v. Furthermore from (4.58), Ω = Ω−1
(4.67)
if acting on G0 . Now at roots of unity, define H to be the algebra U × F with the additional relation
Q = 0, i.e.
H ≡ (U × F )/Q. 89
(4.68)
It is closely related to the H0 representations of section 4.1, see also Conjecture 4.1.3. This is very similar to the definition of the physical Hilbert space in QFT using the
BRST operator (3.8), formulated in terms of an algebra instead of representations, which is essentially the same. Since Q vanishes on all generic representations, this contains in particular the (semidirect product of U with the) physical many–particle
states in the case of SOq (2, 3), as well as the states of the usual truncated tensor product in the context of CFT. Using Ω, we can essentially define an involution on H (resp. for G0 –type repre-
sentations of U × F ) as follows: for x ∈ U, it is simply the involution x corresponding
to the reality structure considered. If x is either Xi± or Hi , this can be written as x = ΩS(x)Ω−1 .
(4.69)
ai = Ωai Ω−1 ,
(4.70)
For a generator ai of F , we define
and extend this as an antilinear antialgebra–homomorphism on H (resp. U × F ). It
is shown in Appendix A that this is consistent with the algebra H provided the ai
are unitary w.r.t. the reality structure on U, and with the braiding algebra (4.61) if
γ is a phase. It is also consistent with (4.60) provided a2 = (−1)fa a2 where (−1)fa is the value of ε on ai ; note that gij∗ = (−1)fa gji, using (4.59). Again in the noncompact case, Ω can act on ai only if it contains both positive and negative energy states, such as H0 in Theorem 4.1.2. Now
√ −2 √ 2 ai = v v −1 εai v −1 v −1 ε−1 = ai (−1)fa
(4.71) (4.72)
in H (resp. on G0 –type representations), using (4.36) resp. (4.66). This is as good as an involution, and the main result of this section. In fact, one could define instead
ai = ǫΩai Ω−1 , which really is an involution; we choose not to do this here. Again, the operator adjoint can be calculated once there is a positive definite inner product. We want to indicate how this could be achieved: Let F be as above for braided copies ai , bi , ... of the algebra (4.60). Define an
evaluation of H
hxf (a, b, ...)i ≡ ǫ(x)hf (a, b, ...)i
(4.73)
by first collecting the generators of the same algebra using the braiding relations, and defining ha2 i ≡ (−1)fa ga with ga ∈ IR, and haj i ≡ 0; similarly for the other variables. 90
This is independent of ordering if γ = ±1, because a2 is then central. It satisfies hxf (a, b, ...)i = hf (a, b, ...)xi = ǫ(x)hf (a, b, ...)i
(4.74)
hf (a, b, ...)i∗ = hf (a, b, ...)i(−1)f .
(4.75)
on H where f = fa + fb + ..., using a2 = a2 (−1)fa .
One can now write states of H0 in the form f i = f ia ib ... aia bib ...i and gi =
g ia ib ...aia bib ...i where aia , bib , . . . are positive–energy (physical) states, and define an inner product as follows: (f, g) ≡ hf gi.
(4.76)
(x ⊲ f, g) = (f, x ⊲ g)
(4.77)
This is hermitian, invariant
and it is shown in Appendix A that it is in fact the same as the inner product defined in (4.29), (ai bj ..., ak bl ...) ≡ hai bj ...ak bl ...i = (ai ⊗ bj ⊗ ...., ak ⊗ bl ⊗ ...)H ,
(4.78)
if the normalization on the rhs is chosen as (ai , aj ) = ga δji (remember that we work in an orthogonal basis). Invariance can be seen easily: (x ⊲ f, g) =
D
E
D
E
D
(x1 f Sx2 )g = Sx2 (f )x1 g = f xg
= (f, x ⊲ g),
E
(4.79)
using (4.74). Hermiticity follows from (4.75). Positivity was discussed in section 4.2.1 for the case of the Anti–de Sitter group. One could formulate all this without using w explicitely, in the form ai = (Ω1 ⊲ ai )Ω2 which can be written in terms of the universal R. Needless to say, this would
be much more complicated. However it helps to understand the main point of this definition, namely the R involved which ”corrects” the flipping of the tensor product
in the reality structure (1.74). In this form, a somewhat similar–looking conjugation was introduced in [40].
4.4.1
On Quantum Fields and Lagrangians
In this section, we want to show how the above formalism could find application in a QFT. This can only be very vague at present, because an important piece is 91
still missing – the implementation of a symmetrization postulate, in order to define identical particles. We can nevertheles write down a few generic formulas in an ad–hoc way. Consider a (large) number of braided copies of the algebra (4.60) with generators λ,(n) ai ,
for n = 1, 2, ..., N and λ going through all possible highest weights of physical
(unitary) representations of a given spin; remember that there exist only finitely many at roots of unity. Then consider the following object: 1 X λ,(n) i ai fλ,(n) (y). Ψ(y) ≡ √ N i,n λ,(n)
i Here fλ,(n) (y) is the dual representation of ai
(4.80)
, realized as functions (or forms, ...)
on quantum AdS space, which is a right U –module algebra in the dual picture of chapter 2. This works as in Theorem 4.1.2, where the factors Vi are the 5–dimensional
representations y i. Then the unitary representations are the quotients H0 , in the space of functions on quantum AdS space. By this construction, Ψ(y) contains both
positive and negative energy representations as discussed earlier, so that Ω can act on it. Thus we can assume that u⊲
X
λ,(n) i fλ,(n) (y)
ai
=
X
λ,(n) jλ i πi (u)fλ,(n) (y)
aj
=
X
λ,(n)
aj
j (y) ⊳ u). (fλ,(n)
(4.81)
for u ∈ U. Ψ(y) behaves very much like an off–shell quantum field. Similarly, we can
consider Ψ(y1), Ψ(y2 ), ... with (braided) copies yi of quantum AdS space. Then for example, hΨ(y1)Ψ(y2 )i = ±
X
j i gλ gij fλ,(n) (y1 )fλ,(n) (y2 ) + o(h)
(4.82)
λ
becomes a correlator in the classical limit, depending on the choice of the gλ ≡ gaλ
for the representations involved. In particular, gλ =
i 2λ −m2
corresponds to a Green’s
function, where 2λ is the (quantum) quadratic Casimir (see [18]). More generally, hΨ(y1 )...Ψ(yk )i
(4.83)
will reproduce the sum of Wick contractions at q = 1 for N → ∞.
Using this, one can write down e.g. an ”interaction Lagrangian” S=
Z
y
Ψ(y)...Ψ(y),
(4.84)
where the integral over quantum AdS space is defined in section 2.3.1 (in different notation), and many similar terms. This is invariant under SOq (2, 3), i.e. xS = Sx 92
in U × F for x ∈ U, using the invariance of the integral. The only thing we want do here is to show that S is in fact hermitian, with the involution defined in the previous section:
Z
y
Ψ(y)...Ψ(y) =
Z
y
Ψ(y)...Ψ(y).
(4.85)
If one can find an algebra replacing (4.60) such that the inner product (4.76) is positive definite, this implies that eiS is a unitary operator on H0 (in the present version, this is not the case because of the additional generator a2 ). λ,(1)
To see (4.85), let us simplify the notation first by writing ai , bj , ck for ai λ,(3)
and ak
λ,(2)
, aj
, respectively, and to be specific consider S˜ =
where ψa (y) ≡
P
i
Z
y
ψa (y)ψb(y)ψc (y)
(4.86)
ai f i(y), and similarly ψb (y), ψc (y). We claim that Z
S˜ =
y
ψc (y)ψb (y)ψa (y).
(4.87)
It is clear that this implies (4.85). We consider only scalar fields here for simplicity. To see (4.87), first observe that b
f i(y) = ±f i (y)
(4.88)
using the (auxiliary) antilinear involution on quantum AdS space defined in section b
2.3.1. This follows from y i = −y i and their algebra, (4.43) and (4.44). Together with
(2.88), this implies Z
y
ψa (y)ψb(y)ψc (y) = =
X
Z
y
Ωck bj ai Ω−1
Z
y
f k (y)f j (y)f i(y)
ψc (y)ψb (y)ψa (y)
(4.89)
as claimed, since this is a scalar and it therefore commutes with U and Ω.
One could now go ahead and define ”ad hoc” correlators such as hΨ(y1)eiS Ψ(y2 )i,
which in fact can be expressed as sum of contractions as seen above, similar to the classical Wick expansion. Moreover for q 6= 1, these contractions can be interpreted
naturally as generalized links, with interaction vertices being Clebsch–Gordan coeffiˆ –matrices in the particular cients defined by the integral, and crossings given by the R
representations. All these diagrams would be finite, since there exist only finitely many ”physical” representations, which are all finite–dimensional. Moreover, these correlators are in fact symmetric in the classical limit, since the Ψ(yi ) commute for 93
q = 1. Nevertheless, this is not really satisfactory, since an explicit symmetrization postulate is missing, as well as a dynamical principle determining ”on–shell” states. These two open problems are probably related. Symmetrization. Let us briefly discuss the problem of symmetrization and identical particles. For generic q, this has been studied in [21]. At roots of unity, the situation can be expected to be somewhat different. In principle, one can define projectors (P S,A )ij kl which act on the tensor product of 2 identical representations V (µ), and project out the totally symmetric resp. ˆ –matrix antisymmetric representations. This can be done using the fact that the R 1
discussed in section 1.2.2 has eigenvalues ±q 2 (cλ −2cµ ) with the same sign as classically, 1
and one ”just” has to correct the q 2 (cλ −2cµ ) –factor. The problem is that this may not give an interesting associative algebra of ”totally (anti)symmetric” representations for more than 2 particles, it is probably too restrictive in general. One may hope that something more favorable happens at roots of unity on the space H0 . In this context, we can define an interesting algebra: ai aj = ±aj ai
(4.90)
with the bar as in (4.70), taking advantage of U × F resp. H. It is easy to check that
this is compatible with the cross–product and our ”involution” on H. Acting on the
Fock–vacuum i, this in fact defines totally symmetric resp. antisymmetric 2–particle √ representations, with a suitable definition of v. Again, it is not clear if this algebra is interesting for our purpose for more than 2 particles. It would be very desirable to get a better understanding of these issues.
4.4.2
Nonabelian Gauge Fields from Quantum AdS Space
In section 3.2.6, we have found a BRST operator for spin one particles, corresponding to abelian gauge fields. Needless to say that one would also like to consider the nonabelian case. Nonabelian gauge fields are usually described by connections on principal fiber bundles. Therefore one might try to do the same in the q–deformed case, see e.g. [4, 46]. However there is a much simpler, extremely fascinating way to obtain such objects on quantum AdS space (and certain other quantum spaces). It is in fact much simpler than classically. We will only give a rough outline here. Consider the calculus of differential forms on q–AdS space, which is the same as on quantum Euclidean space, except for the reality structure. As in 2.2.4, the 94
following observation by Bruno Zumino [65] will be crucial: there exists a ”radial” one–form ω =
√
q 1 √ d(r 2) q −1 + q r 2
which generates the calculus on quantum Euclidean
space, sphere resp. on q-AdS space by √ √ [ω, f ]± = ( q−1 − q)df ≡ ξdf.
(4.91)
Furthermore on the sphere, it is not possible for q 6= 1 to work with ”tangential” one– forms only, due to the commutation relations (2.15). Therefore ω must be included in the calculus. Now consider a matrix of one–forms Ωij (this has nothing to do with the Ω in section 4.4), and write it in the following way: Ωij = ωδji + ξBji .
(4.92)
Physically, we can imagine that this comes from some spontaneous symmetry breaking in the radial fields, which are scalars. One can decompose the one-forms Bji into tangential and radial components, e.g. using a Hodge–star operator ∗ which can be
defined in a straightforward way. Then
Ω = ω1 + ξ(Φ + A),
i.e.
Ωij = ωδji + ξ(φij ω + Aijµ dy µ )
(4.93)
with the condition ω ∧ ∗(Aij,µdy µ ) = 0, changing notation for the coordinates on AdS space. Furthermore, we can imagine a reality condition like (Ωij )† = −Ωji (without
going into details here), so that Ωij corresponds to some Lie algebra, and (4.92) corresponds to trφ = 0. Now consider (Ω2 )ij , which after a simple calculation using (4.91) becomes Ω2 = ξ 2 (dB + BB),
i.e.
Ωik Ωkj = ξ 2 (dBji + Bki Bjk ).
(4.94)
Decomposing it into radial and tangential components, we get Ω2 = ξ 2 (dA + AA + dΦ + ΦA + AΦ) = ξ 2 (dA + AA + (dφ + [A, φ])ω + o(h))
(4.95)
where q = e2πih as usual. Thus S=
1 Z tr(Ω2 ∗ Ω2 ) 2 ξ 95
(4.96)
gives precisely the Yang–Mills action for a gauge field A coupled to a scalar in the adjoint, like a Higgs field in some GUT models! We do not have to define curvature by hand. This also contains massless BRST ghosts in a nonstandard form, according to section 3.2.6. Moreover, if Ω transforms like Ω(y) → γ −1 (y)Ω(y)γ(y),
(4.97)
then the components of Ω transform like A → γ −1 Aγ + γ −1 dγ + o(h), φ → γ −1 φγ + o(h),
and (4.98)
which are precisely the gauge transformations in a Yang–Mills Theory. Similarly, any trace of polynomials in Ω gives a gauge invariant Lagrangian in the limit h → 0.
How exactly this fits together with the BRST operator etc. remains to be seen.
At the very least, this shows that there is no need to define objects like connections on principal bundles for q 6= 1, they arise naturally form the mathematical structure
considered. Without elaborating this any further here, we see once again that q –
deformation is more than just a ”deformation”, it allows to do things which cannot be done for q = 1, and which look very interesting from the point of view of QFT.
96
Chapter 5 Conclusion We have shown that many of the essential ingredients of quantum field theories, or more properly quantum theories of elementary particles, have their counterparts in an approach where the classical Poincare group and Minkowski–space are replaced by the quantum Anti–de Sitter group SOq (2, 3) and a corresponding AdS space, with q a suitable root of unity. First of all, it turned out that there are indeed unitary representations which may describe elementary particles of any spin, with the same low–energy structure as classically. We have defined many–particle representations, which are unitary as well; this is not trivial. In the massless case, it turns out that there is a very natural way to define a Hilbert space using a BRST operator Q, which is an element of the center of Uq (so(2, 3)); in the classical case, it has to be defined by hand. Moreover, the same Q works for any spin, and it seems that it can be used to define the many–particle Hilbert space in a similar way. We have also seen how the structure of a nonabelian gauge theory can arise naturally from quantum Anti–de Sitter space; again in the classical case, its description using connections on fiber bundles is quite ad–hoc. Moreover, everything is manifestly finite in the quantum case. This should not be considered a technicality; it seems that if there is a complete theory of elementary particles, it should be possible to formulate it in a completely well–defined way, without any vague ”remedies” under the name of regularization; it should be regular by itself. While it may be too ambitious to attempt finding such a theory, any finite version of a 4–dimensional quantum field theory would be very interesting in itself. Apart from all these mathematical features, one may still wonder if it makes any sense at all to assume that the spacetime we see looking out of the window is noncommutative, i.e. not a classical manifold. I think that this has been answered as 97
well, by pointing out that if q = eiπh is very close to 1 (which it has to be, otherwise there is an obviously clash with observation), the coordinate algebra of quantum √ Anti–de Sitter space is classical up to corrections involving a length scale L0 = hR, where R is the curvature radius of AdS space. Thus one should expect it to look like a classical manifold on length scales bigger than L0 , just like quantum mechanics behaves classically on scales large compared to h ¯ . This shows that h must indeed be a very small number. All this can be said in favour of the approach chosen. Nevertheless, we have not yet succeeded in formulating a theory of elementary particles, because we do not know at this time how to define a Hilbert space of identical particles. There may be a satisfactory solution of this problem in the form of a suitable algebra of creation and anihilation operators (in which case we have presented an nice way to define an involution, which is usually a difficult part in the noncommutative case), or perhaps in a completely unexpected way, – or it might be that this is the downfall of the approach. Of course one should hope for the first alternatives, and in view of all the promising features found so far, I cannot believe that this is the end of the story.
98
Appendix A Consistency of Involution, Inner Product Compatibility of (4.70) with cross product. Applying (.) to xai = x1 ⊲ ai x2 in H, we get Ωai Ω−1 x = (x2 )al πil (x1 )∗ = x2 Ωal Ω−1 πli (x1 ).
(A.1)
Multiplying from the left with Ω−1 and from the right with Ω, this becomes !
ai Ω−1 xΩ = Ω−1 (x)1 Ωal πli ((x)2 )
(A.2)
where (x)1,2 is the Sweedler notation for the coproduct, which becomes using (4.39) !
e −1 x = (θS e −1 (x) ) ⊲ a (θS e −1 (x) ) π i ((x) ) ai θS 1 1 l 1 2 l 2 e −1 (x) ) ⊲ a θS e −1 (x) π i ((x) ) = (θS 12 l 11 l 2
e −1 (x) = ak πkl (S −1 (x)12 )πli ((x)2 )θS 11 e −1 (x) = ak πki ((x)2 S −1 (x)12 )θS 11
e −1 (x) = ak πki (ǫ(x2 ))θS 1 e −1 (x) = ai θS
(A.3)
as desired, using Lemma 4.3.1 and standard properties of Hopf algebras. Compatibility of (4.70) with braiding. Applying (.) to ai bj = γ(R2 ⊲ bj )(R1 ⊲ ai ), we get !
Ωbj ai Ω−1 = γ ∗ Ωak πik (R1 )∗ bl πjl (R2 )∗ Ω−1 , 99
(A.4)
or !
bj ai = γ ∗ ak πki (R1 )bl πlj (R2 ) j −1 = γ ∗ ak πki (R−1 2 )bl πl (R1 ) l −1 = γ ∗ ak πik (R−1 1 )bl πj (R2 ),
(A.5)
using (θ ⊗ θ)R = R21 and similarly for θe in the last line. Multiplying with πsj (Rb )πti (Ra ),
we get
!
bj πsj (Rb )ai πti (Ra ) = γ ∗ at bs ,
(A.6)
which is just the braiding algebra provided γ ∗ = γ −1 . √ Equality of the inner products (4.78) Using ∆(Ω) = ∆( v −1 )R−1 (w˜ ⊗ w) ˜ and
hf (a)x = h(S −1 x) ⊲ f (a) we can write the lhs of (4.78) explicitely as √ √ √ ˜ ⊲ (...bj ai )( v −1 )2 R−1 ˜ w˜ −1 vak bl ...i (ai bj ..., ak bl ...) = h(( v −1 )1 R−1 1 w) 2 w √ √ √ ˜ ⊲ (...bj ai )( v −1 )2 R−1 vak bl ...i = h(( v −1 )1 R−1 1 w) 2 √ √ √ = h(S −1 (( v −1 )2 R−1 v)( v −1 )1 R−1 ˜ ⊲ (...bj ai )ak bl ...i 2 1 w) √ √ = h(S −1 (R−1 v)ǫ( v −1 )R−1 ˜ ⊲ (...bj ai )ak bl ...i 2 1 w) √ = h( vR2 S 2 R1 w) ˜ ⊲ (...bj ai )ak bl ...i √ −2˜ρ −1 = h( vq v w) ˜ ⊲ (...bj ai )ak bl ...i √ = h( v(−1)E v −1 w) ⊲ (...bj ai )ak bl ...i √ = h(v −1 w) ⊲ (...bj ai ) v(−1)E ak bl ...i (A.7) √ using (1.29) ff., (1.54), (4.38) and h(x ⊲ a)bi = ha(Sx)bi, ǫ( v −1 ) = 1, which is easy to see. In particular, with (4.52) we can verify that √ (ai , aj ) ≡ hai aj i = h(v −1w ⊲ ai ) v(−1)E aj i 1
= ga q 2 ca (−1)Ej (−1)fa gkj πik (w) = ga q ca (−1)Ej +fa πkj ((−1)E w)πik (w) = ga δji
(A.8)
is a (positive definite) inner product on the unitary representations ai ; it had to come out this way, since we always use the orthogonal basis of Lemma 4.3.1. Using the first line of (A.8) and ∆(n) (v −1 w) = (v −1 w ⊗ ... ⊗ v −1 w)R(n) , we can
continue (A.7) as
√ rhs = h...(v −1 w) ⊲ bs πjs (Rn−1 )(v −1 w) ⊲ at πit (Rn ) v(−1)E ak bl ...i √ 1 = q 2 (ca +cb +...)πit (Rn )πjs (Rn−1 )...(at ⊗ bs ⊗ ..., v ⊲ (ak ⊗ bl ⊗ ...))⊗ (A.9) 100
where R12... are the components of R(n) assuming there are n factors, and using ∆(−1)E = (−1)E ⊗ ... ⊗(−1)E . Now notice that (A.8) and (1.63) imply πit (R2 )πjs (R1 )(at ⊗ bs , ak ⊗ bl )⊗ = ga gb πik (R2 )πjl (R1 )
= (ai ⊗ bj , at ⊗ bs )⊗ πkt (R1 )πls (R2 )
= (ai ⊗ bj , ak ⊗ bl )R
(A.10)
and similarly for several factors, so (A.9) is nothing but rhs = cV (ai ⊗ bj ⊗ ....,
√
v ⊲ (ak ⊗ bl ⊗ ...))R
= (ai ⊗ bj ⊗ ...., ak ⊗ bl ⊗ ...)H as in (4.29).
101
(A.11)
Appendix B On the Weyl Element w In this appendix, we want to give some explanations to the remarkable formulas (4.34), (4.35) and (4.37). As mentioned before, the braid group action (1.46) on U can in fact be extended
to a braid group action Ti on any representation, with explicit formulas in terms of (infinite) sums of generators of U [39], see also [25]. Then one can consider the
element of the braid group corresponding to the longest element of the Weyl group, call it T . It was shown in [33, 34] that these Ti and therefore T can be implemented by a conjugation with wi resp. w, which are elements of an extension of U, and that
w satisfies (4.34) with suitable definitions. Unfortunately, this requires complicated calculations. To get some faith in these formulas, we want to point out first that if (4.34) ∆(w) = R−1 w ⊗ w
(B.1)
holds on the tensor product of 2 fundamental representations, it holds for any representations. This can be seen inductively: if (B.1) holds on Vi ⊗ Vj , then it also holds on the representations in V1 ⊗ V2 ⊗ V3 , since the rhs of !
(id ⊗ ∆)∆(w) = R−1 1,(23) (w ⊗ w)
(B.2)
acting on V1 ⊗(V2 ⊗ V3 ) agrees with the rhs of !
(∆ ⊗ id)∆(w) = R−1 (12),3 (w ⊗ w)
(B.3)
acting on (V1 ⊗ V2 ) ⊗ V3 , by the first statement of Lemma 1.1.1. Thus the lhs agree as they should.
102
Furthermore, since the action of w on any given (finite–dimensional) representation can be expressed in terms of generators of U, it satisfies ∆′ (w) = R∆(w)R−1 ,
and together with (B.1) this implies ∆(w) = w ⊗ wR21 .
The action of wi resp. w on the fundamental representations can be found ex-
plicitely, see [49]. For real groups, it is essentially the invariant metric gij . (4.35) together with the statement that ε = ±1 on any irrep now follows easily,
since ε ≡ v −1 w 2 is grouplike as already pointed out, i.e. ∆(ε) = ε ⊗ ε, and it only
remains to check that ε is ±1 on the fundamental representations, where it agrees
with the classical limit. This also follows from consistency with the tensor product.
A proof of wxw −1 = θSx for real groups. Here we only consider the case of ”real” groups, which are all except An , Dn and E6 , i.e. those for which the Dynkin diagram does not have an automorphism. From (4.34), (1.49) and (1.29), we know that (w ⊗ w)R(w −1 ⊗ w −1 ) = R21 = (θS ⊗ θS)R
(B.4)
Furthermore, it is known that the first terms of the expansion of R in powers of Xi± are
R=q
P
(α−1 )ij hi ⊗ hj
1+
X
ci (q)q
1 h 2 i
Xi+
i
⊗q
− 12 hi
Xi−
!
+ ...
(B.5)
Now wXi± w −1 certainly has the form U 0 Xi∓ , as can be seen either from the explicit
formulas in [33, 34] or from the cross product algebra (4.62). On the Cartan subalgebra, w acts as classically. Thus by the uniqueness of R [32, 16], (B.4) implies that
wXi± w −1 = ai θS(Xi∓ )
(B.6)
∓ wXi± w −1 = a−1 i θS(Xi )
(B.7)
and
with a constant ai . Notice that if there exists an automorphism of the Dynkin diagram, then this would only follow up to a corresponding permutation of the simple root vectors. The constants ai can be eliminated by a redefinition w → wq bi Hi with
suitable constants bi , using the fact that the Cartan matrix is non–singular. This new w now satisfies all the equations discussed.
103
Bibliography [1] L. Alvarez–Gaum´e, C. Gomez, G. Sierra, ”Duality and Quantum Groups” Nucl. Phys. B330, 347 (1990) [2] H.H. Anderson, P. Polo, W. Kexin, ”Representations of Quantum Algebras” Invent. math. 104, 1 (1991) [3] T. Banks, W. Fischler, S. H. Shenker, L. Susskind, ”M Theory As A Matrix Model: A Conjecture” Phys.Rev. D55, 5112 (1997); P–M. Ho, Y–S. Wu, ”Noncommutative Geometry and D–Branes” hep-th/9611233 [4] T. Brzezinski and S. Majid, ”Quantum group gauge theory on quantum spaces” Comm. Math. Phys. 157, 591 (1993); C.-S. Chu, P.-M. Ho and H. Steinacker, ”q–deformed monopole with arbitrary charge” Z. Phys. C 71, 171 (1996) [5] A. Chamseddine, G. Felder, J. Fr¨ohlich, ”Gravity in Non-Commutative Geometry” Comm. Math. Phys. 155, 205 (1993) [6] V. Chari and A. Pressley, ”A guide to quantum groups”. Cambridge University press, 1994 [7] C. Chryssomalakos and B. Zumino, ”Translations, Integrals and Fourier Transforms in the Quantum Plane” preprint LBL-34803, UCB-PTH-93/30, in A. Ali, J. Ellis and S. Randjbar-Daemi (eds.), Salamfestschrift, World Scientific, 1993 [8] A. Connes, ”G´eom´etrie non commutative”, Publ. Math. IHES Vol. 62 41 (1986) [9] A. Connes, In: The interface of mathematics and particle physics, D. Quillen, G. Segal, S. Tsou (eds.), Oxford: Clarendon 1990 [10] A. Connes, J. Lott, ”Particle models and noncommutative geometry” Nucl. Phys. B (Proc. Suppl.) 18B, 29 (1990) 104
[11] M. Dijkhuizen and T. Koornwinder, ”CQG Algebras: A Direct Algebraic Approach to Compact Quantum Groups” Preprint, hep-th 9406042 [12] L. Dabrovski, V. K. Dovrev, R. Floreanini, V. Husain, Phys. Lett 302B, 215 (1993); V. K. Dobrev, P. J. Moyan, Phys. Lett. 315B, 292 (1993); V. K. Dobrev, R. Floreanini, J. Phys. A: Math. Gen. 27, 4831 (1994); V. K. Dobrev, Invited talk at the Workshop on Generalized Symmetries in Physics (Clausthal, July 1993), Proceedings, eds. H.-D. Doebner, V. K. Dobrev, A. G. Ushveridze (World Sci., Singapore, 1994), p. 90 [13] C. De Concini, V. G. Kac, ”Representations of Quantum Groups at Roots of 1” Operato Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, A. Connes et al. (eds), p. 471, Birkhaeuser, Boston [14] P. Dirac, J. Math. Phys. 4, 901 (1963); M. Flato, C. Fronsdal, Lett. Math. Phys 2, 421 (1978) [15] V. Drinfeld, ”Quantum Groups” Proceedings of the International Congress of Mathematicians, Berkeley, 1986 A.M. Gleason (ed.), p. 798, AMS, Providence, RI [16] V. Drinfeld, ”On Almost Cocommutative Hopf Algebras” Leningrad Math. J. 1, No. 2, 321 (1990) [17] V. Drinfeld, ”Quasi–Hopf Algebras” Leningrad Math. J. 1, No. 6, 1419 (1990) [18] L.D.Faddeev, N.Yu.Reshetikhin, L.A.Takhtajan. ”Quantization of Lie Groups and Lie Algebras” Algebra Anal. 1(1987)178 [19] G. Fiore, ”The SOq (N) - Symmetric Harmonic Oscillator on the Quantum Euclidean Space RqN and its Hilbert Space Structure” Int. J. Mod. Phys 26, 4678 (1993) Groups and Lie Algebras” Algebra Anal. 1 178 (1989) [20] G. Fiore, Ph.D - thesis, May 1994 [21] G. Fiore, P. Schupp, ”Identical Particles and Quantum Symmetries” hepth/9508047; G. Fiore, ”Deforming Maps from Classical to Quantum Group Covariant Creation and Annihilation Operators” q-alg/9610005
105
[22] C. Fronsdal, Rev. Mod. Phys. 37, Nr. 1, 221 (1965); C. Fronsdal, Phys. Rev. D 10, Nr. 2, 589 (1974); C. Fronsdal, R. Haugen, Phys. Rev. D 12, Nr. 12, 3810 (1975); C. Fronsdal, R. Haugen, Phys. Rev. D 12, Nr. 12, 3819 (1975); C. Fronsdal, Phys. Rev. D 20, 848 (1979); J. Fang, C. Fronsdal, Phys. Rev D 22, 1361 (1980) [23] J. Fuchs, ”Affine Lie Algebras and Quantum Groups” Cambridge Monographs on Mathematical Physics, Cambridge University Press 1992 [24] A. Hebecker and W. Weich, ”Free Particle in q - deformed Configuration space”. Lett. Math. Phys. 26, 245 (1992). [25] J.C. Jantzen, ”Lectures on Quantum Groups”. Graduate Studies in Mathematics Vol. 6, AMS 1996 [26] J.C. Jantzen, ”Moduln mit h¨ochstem Gewicht”, Lecture notes in Mathematics, Springer, vol. 750; V. Kac, D. Kazhdan, ”Structure of Representations with Highest Weight of infinite-dimensional Lie Algebras”, Adv. Math. 34, 97 (1979) [27] M. Jimbo, ”A q – Difference Analogue of U(g) and the Yang – Baxter Equation” Lett. Math. Phys 10, 63 (1985) [28] B. Jurco, private communication [29] T. Kato, ”Perturbation Theory for linear Operators”. Springer (Berlin, Heidelberg, New York) 1980, 2nd edition | )), q m = 1”. Lett. Math. Phys. 21, 273 [30] G. Keller, ”Fusion Rules of Uq (sl(2, C
(1991) [31] A. Kempf and S. Majid, ”Algebraic q - integration and Fourier theory on quantum and braided spaces” J. Math. Phys. 35, 6802 (1994) [32] S.M. Khoroshkin, V.N. Tolstoy, ”Universal R- Matrix for Quantized (Super)Algebras” Comm. Math. Phys 141, 599 (1991) [33] A.N. Kirillov, N. Reshetikhin, ”q- Weyl group and a Multiplicative Formula for Universal R- Matrices” Comm. Math. Phys. 134, 421 (1990) [34] S.Z. Levendorskii, Y.S. Soibelman, ”Some applications of the quantum Weyl group” Journ. Geom. Phys 7 (2), 241 (1990) 106
[35] J. Lukierski, A. Novicki, H. Ruegg, ”Real forms of complex quantum Anti- de| )) and their contraction schemes” Phys. Lett. B 271, Sitter algebra Uq (Sp(4; C
321 (1991) [36] J. Lukierski, H. Ruegg, A. Nowicki, V. Tolstoy, ”q – deformation of Poincare algebra” Phys. Lett. B 264, 331 (1991); J. Lukierski, A. Nowicki, H. Ruegg, ”New quantum Poincare algebra and κ – deformed field theory”, Phys. Lett. B 293, 344 (1992); [37] G. Lusztig, ”Quantum deformations of certain simple modules over enveloping algebras”, Adv. in Math. 70, 237 (1988); G. Lusztig, ”On quantum groups” J. Algebra 131, 466 (1990) [38] G. Lusztig, ”Quantum Groups at roots of 1” Geom. Ded. 35, 89 (1990) [39] G. Lusztig, ”Introduction to Quantum Groups”. Progress in Mathematics Vol. 110, Birkhaeuser 1993 [40] G. Mack, V. Schomerus, ”Quasi Hopf quantum symmetry in quantum theory” Nucl. Phys. B 370, 185 (1992) [41] S. Majid, ”Quasitriangular Hopf algebras and Yang-Baxter equations”, Int. J. Mod. Phys. A 5, 1 (1990) [42] S. Majid, ”Beyond Supersymmetry and Quantum Symmetry”. In M-L. Ge and H.J. deVega, editors, Quantum Groups, Integrable Statistical Models and Knot Theory, p. 231, World Sci., 1993. [43] U. Meyer, ”Quantum determinants”. DAMTP/94-54, hep-th/9406172 [44] O. Ogievezky and B. Zumino, ”Reality in the Differential Calculus on q - Euclidean Spaces.” Lett. Math. Phys. 25, 121 (1992) [45] V. Pasquier, H. Saleur, ”Common Structures between Finite Systems and Conformal Field Theories through Quantum Groups” Nucl. Phys. B 330, 523 (1990) [46] M. Pflaum, ”Quantum groups on Fibre Bundles” Comm. Math. Phys. (1994) [47] O. Ogievetsky, W. Schmidke, J. Wess, B. Zumino, Comm. Math. Phys. 150, 495 (1992); P. Aschieri, L. Castellani, Lett. Math. Phys 36, 197 (1996); M.
107
Chaichian, A. Demichev, Phys. Lett. B 304, 220 (1993); P. Podles. S. Woronowicz, hep-th/9412059, UCB preprint PAM-632 V.K. Dobrev, J. Phys. A: Math. Gen. 26, 1317 (1993) [48] P. Podles, ”Compact Quantum Groups and their real Representations”, Publ. RIMS, Kyoto Univ., 28 (1992), 709 [49] N. Reshetikhin, ”Quantized universal enveloping algebras, the Yang–Baxter equation and invariants of links I,II” LOMI preprint E-4-87, E-17-87. [50] M. Rosso, ”Finite Dimensional Representations of the Quantum Analog of the Enveloping Algebra of a Complex Simple Lie Algebra” Comm. Math. Phys. 117, 581 (1988) [51] P. Schupp, ”Quantum Groups, Non–Commutative Differential Geometry and Applications” Ph.D. thesis Berkeley, 1993. hep-th 9312075 [52] P. Schupp, P. Watts and B. Zumino, ”Differential Geometry on Linear Quantum Groups” Lett. Math. Phys. 25, 139 (1992) [53] Y.S. Soibelman, ”Quantum Weyl group and some of its applications” Rend. Circ. Mat. Palermo Suppl 26 (2), 233 (1991) [54] M. E. Sweedler, Hopf Algebras, (Benjamin, 1969) [55] H. Steinacker, ”Integration on quantum Euclidean space and sphere” J. Math. Phys. 37 (Nr.9), 7438 (1996) [56] H. Steinacker, ”Finite – dimensional Unitary Representations of quantum Anti – de Sitter Groups at Roots of Unity” q-alg/9611009 [57] P. Watts, ”Differential Geometry on Hopf Algebras and Quantum Groups” Ph.D. thesis Berkeley, 1993. hep-th 9412153 [58] J. Wess, B. Zumino, ”Covariant Differential Calculus on the Quantum Hyperplane” Nucl Phys 18B (Proc. Suppl.), 302 (1990) [59] H. Weyl, ”The classical groups”. Princeton University press 1946 Pseudogroups”. Comm. Math. Phys. 122, 125 - 170 (1989) [60] E. Witten, ”Bound States Of Strings And p-Branes” Nucl.Phys. B 460, 335 (1996) 108
[61] S. L. Woronowicz, ”Compact Matrix Pseudogroups”, Commun. Math. Phys. 111 613 (1987) [62] S.L. Woronowicz, ”Differential Calculus on Compact Matrix Pseudogroups”. Comm. Math. Phys. 122, 125 (1989) [63] B. Zumino, Nucl. Phys. B 127, 189 (1977); S. Deser, B. Zumino, Phys. Rev. Lett 38, 1433 (1977) [64] B. Zumino, ”Introduction to the Differential Geometry of Quantum Groups”, In: K. Schm¨ udgen (Ed.), Math. Phys. X, Proc. X-th IAMP Conf. Leipzig (1991), Springer-Verlag (1992) [65] B. Zumino, private communication
109
